Class 10 maths Introduction to Trigonometry Hots Questions

Q 1 :

In $∆ A B C$ right angled at B, $\sin A = \frac{7}{25},$ then the value of cos C is ............................

$\frac{7}{25}$
$\frac{24}{25}$
$\frac{7}{24}$
$\frac{24}{7}$

1

2

3

4

(1) $\frac{7}{25}$

Q 2 :

If 5 tan θ = 4, then the value of $\frac{5 \sin θ - 3 \cos θ}{5 \sin θ + 2 \cos θ}$ is

1/6
1/7
1/4
1/5

1

2

3

4

(1) 1/6

Q 3 :

Given that sin α = 1/2 and cos β = 1/2, then the value of (β – α) is

0°
30°
60°
90°

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2

3

4

(2) 30°

Q 4 :

If sinθ + cosθ = $\sqrt{2}$ , then tanθ + cot θ =

1
2
3
4

1

2

3

4

(2)

$\sin θ + \cos θ = \sqrt{2}$

Squaring on both sides, we get

$\sin^{2} θ + \cos^{2} θ + 2 \sin θ \cos θ = 2$

$\Rightarrow 1 + 2 \sin θ \cos θ = 2$ $[∵ \sin^{2} θ + \cos^{2} θ = 1]$

$\Rightarrow 2 \sin θ \cos θ = 1 \Rightarrow \sin θ \cos θ = \frac{1}{2}$

But $\tan θ + \cot θ = \frac{\sin θ}{\cos θ} + \frac{\cos θ}{\sin θ} = \frac{\sin^{2} θ + \cos^{2} θ}{\cos θ \sin θ} = 2$

Q 5 :

If 5 tanβ = 4, then $\frac{5 \sin β - 2 \cos β}{5 \sin β + 2 \cos β} =$

1/3
2/5
3/5
6

1

2

3

4

(1)

$5 \tan β = 4 \Rightarrow \tan β = \frac{4}{5}$

$∴$ $\frac{5 \sin β - 2 \cos β}{5 \sin β + 2 \cos β} = \frac{5 \tan β - 2}{5 \tan β + 2}$ $[dividing \cos β by Nr. and Dr.]$

$= \frac{5 \times \frac{4}{5} - 2}{5 \times \frac{4}{5} + 2} = \frac{2}{6} = \frac{1}{3}$

Q 6 :

If x tan 60°cos 60°= sin60°cot 60°, then x =

cos30°
tan30°
sin30°
cot30°

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2

3

4

(2)

$x \tan 60 ° \cos 60 ° = \sin 60 ° \cot 60 °$

$\Rightarrow x \times \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{3}} \Rightarrow x \sqrt{3} = 1$

$\Rightarrow x = \frac{1}{\sqrt{3}} \Rightarrow x = \tan 30 °$ $[∵ \tan 30 ° = \frac{1}{\sqrt{3}}]$

Q 7 :

If sec θ – tan θ = m, then the value of sec θ + tan θ is :

$1 - \frac{1}{m}$
$m^{2} - 1$
$\frac{1}{m}$
$- m$

1

2

3

4

(3)

Given, $\sec θ - \tan θ = m$ ...(i)

We know that, $\sec^{2} θ - \tan^{2} θ = 1$

$\Rightarrow (\sec θ - \tan θ) (\sec θ + \tan θ) = 1$

$\Rightarrow m (\sec θ + \tan θ) = 1$

$\Rightarrow \sec θ + \tan θ = \frac{1}{m}$

Q 8 :

If cos $(α + β)$ = 0, then value of cos $(\frac{α + β}{2})$ is equal to:

$\frac{1}{\sqrt{2}}$
$\frac{1}{2}$
$0$
$\sqrt{2}$

1

2

3

4

(1)

$\cos (α + β) = 0 = \cos 90 ° \Rightarrow (α + β) = 90 ° \Rightarrow \frac{α + β}{2} = 45 °$

$\Rightarrow \cos (\frac{α + β}{2}) = \cos 45 ° = \frac{1}{\sqrt{2}}$

Q 9 :

The value of $(\sin^{2} θ + \frac{1}{1 + \tan^{2} θ})$ is:

0
2
1
– 1

1

2

3

4

(3) 1

Q 10 :

If $\tan^{2} θ + \cot^{2} α = 2,$ where $θ = 45 ° and θ ° \leq α \leq 90 °,$ then the value of a is :

30°
45°
60°
90°

1

2

3

4

(2) 45°

Q 11 :

If $\sin θ = \cos θ,$ $(0 ° < θ < 90 °),$ then value of $(\sec θ . \sin θ)$ is:

$1 / \sqrt{2}$
$\sqrt{2}$
$1$
$0$

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2

3

4

(3)

$\sin θ = \cos θ \Rightarrow \tan θ = 1 = 45 ° \Rightarrow θ = 45 °$

Now, $\sec θ . \sin θ = \sec 45 ° . \sin 45 ° = \sqrt{2} \times \frac{1}{\sqrt{2}} = 1$

Q 12 :

If $5 t a n θ - 12 = 0$ , then the value of $\sin θ$ is

$\frac{5}{12}$
$\frac{12}{13}$
$\frac{5}{13}$
$\frac{12}{5}$

1

2

3

4

(2)

$G i v e n, 5 t a n θ - 12 = 0 \Rightarrow t a n θ = \frac{12}{5} = \frac{p}{b} L e t p = 12 k, b = 5 k h = \sqrt{p^{2} + b^{2}} = \sqrt{(12 k)^{2} + (5 k)^{2}} = \sqrt{144 k^{2} + 25 k^{2}} = \sqrt{169 k^{2}} = 13 k si n θ = \frac{p}{h} = \frac{12 k}{13 k} = \frac{12}{13}$

Q 13 :

In $∆ A B C$ , right angled at B $s i n A = \frac{7}{25}$ , then the value of cos C is

$\frac{7}{25}$
$\frac{24}{25}$
$\frac{7}{24}$
$\frac{24}{7}$

1

2

3

4

(1)

$s i n A = \frac{p}{h} = \frac{B C}{A C} = \frac{7}{25} c o s C = \frac{b}{h} = \frac{B C}{A C} = \frac{7}{25}$

Q 14 :

In the given figure, D is the mid-point of BC , then the value of $\frac{\cot y}{\cot x} i s$

2
1/2
1/3
1/4

1

2

3

4

(2)

$F r o m t h e g i v e n f i g u r e \frac{\cot y}{\cot x} = \frac{A C}{B C} \div \frac{A C}{C D} = \frac{A C}{B C} \times \frac{C D}{A C} = \frac{C D}{B C} = \frac{C D}{2 C D} = \frac{1}{2} (U s i n g c o t θ = \frac{b}{p})$

Q 15 :

If $a t a n θ = b$ the value of $\frac{b \sin θ - a \cos θ}{b \sin θ + a \cos θ}$ is

$\frac{b - a}{b^{2} + a^{2}}$
$\frac{b + a}{b^{2} + a^{2}}$
$\frac{b^{2} + a^{2}}{b^{2} - a^{2}}$
$\frac{b^{2} - a^{2}}{b^{2} + a^{2}}$

1

2

3

4

(4)

$W e h a v e, t a n θ = \frac{b}{a} G i v e n e x p r e s s i o n i s \frac{b \sin θ - a \cos θ}{b \sin θ + a \cos θ} T a k i n g \cos θ c o m m o n f r o m t h e e x p r e s s i o n, i t g i v e s \frac{b \tan θ - a}{b \tan θ + a} \dots (i) S u b s t i t u t i n g t a n θ = \frac{b}{a} i n e q u a t i o n (i) g i v e s \frac{\frac{b^{2}}{a} - a}{\frac{b^{2}}{a} + a} = \frac{b^{2} - a^{2}}{b^{2} + a^{2}}$

Q 16 :

Which of the following trigonometric ratios are correctly written in context of $∆ A B C a s g i v e n ?$

$(i) s i n A = \frac{B C}{A C} (i i) t a n C = \frac{B C}{A B} (i i i) c o s C = \frac{A B}{A C} (i v) s e c A = \frac{A C}{A B}$

(i) and (ii)
(ii) and (iv)
(i) and (iv)
(ii) and (iii)

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2

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4

(3)

$W h e n ∠ A i s u n d e r c o n s i d e r a t i o n B a s e = A B, P e r p e n d i c u l a r = B C, H y p o t e n u s e = A C ∴ s i n A = \frac{Perpendicular}{Hypotenuse} = \frac{B C}{A C} and s e c A = \frac{Hypotenuse}{Base} = \frac{A C}{A B} W h e n ∠ C i s u n d e r c o n s i d e r a t i o n : B a s e = B C, P e r p e n d i c u l a r = A B, H y p o t e n u s e = A C ∴ t a n C = \frac{Perpendicular}{Base} = \frac{A B}{B C} and c o s C = \frac{Base}{Hypotenuse} = \frac{B C}{A C}$

Q 17 :

If $s i n α = \frac{\sqrt{3}}{2}, c o s β = \frac{\sqrt{3}}{2}, t h e n t a n α \cdot t a n β i s :$

$\sqrt{3}$
$\frac{1}{\sqrt{3}}$
1
0

1

2

3

4

(3)

Given

$s i n α = \frac{\sqrt{3}}{2} = s i n 60 ° \Rightarrow α = 60 ° A n d, c o s β = \frac{\sqrt{3}}{2} = c o s 30 ° \Rightarrow β = 30 ° ∴ t a n α \cdot t a n β = t a n 60 ° \cdot t a n 30 ° = \sqrt{3} \times \frac{1}{\sqrt{3}} = 1$

Q 18 :

The value of $θ$ for which $2 {s i n}^{2} θ = \frac{1}{2}, 0 ° \leq θ \leq 90 ° i s$

$30 °$
$60 °$
$45 °$
$90 °$

1

2

3

4

(2)

$2 {s i n}^{2} θ = \frac{1}{2} \Rightarrow {s i n}^{2} θ = \frac{1}{4} \Rightarrow s i n θ = \frac{1}{2} = s i n 30 ° \Rightarrow θ = 30 °$

Q 19 :

$[\frac{5}{8} {s e c}^{2} 60 ° - {t a n}^{2} 60 ° + {c o s}^{2} 45 °]$ is equal to

$- \frac{5}{3}$
$- \frac{1}{2}$
0
$\frac{1}{4}$

1

2

3

4

(3)

We have,

$[\frac{5}{8} {s e c}^{2} 60 ° - {t a n}^{2} 60 ° + {c o s}^{2} 45 °] = \frac{5}{8} \times (2)^{2} - (\sqrt{3})^{2} + {(\frac{1}{\sqrt{2}})}^{2} = \frac{5}{8} \times 4 - 3 + \frac{1}{2} = \frac{5}{2} - 3 + \frac{1}{2} = \frac{5 - 6 + 1}{2} = 0$

Q 20 :

$\frac{2 t a n 30 °}{1 + {t a n}^{2} 30 °}$ is equal to

$\sin 60 °$
$\cos 60 °$
$\tan 60 °$
$\sin 30 °$

1

2

3

4

(1)

We have,

$\frac{2 t a n 30 °}{1 + {t a n}^{2} 30 °} = \frac{2 \times \frac{1}{\sqrt{3}}}{1 + {(\frac{1}{\sqrt{3}})}^{2}} = \frac{\frac{2}{\sqrt{3}}}{1 + \frac{1}{3}} = \frac{\frac{2}{\sqrt{3}}}{\frac{4}{3}} = \frac{2}{\sqrt{3}} \times \frac{3}{4} = \frac{\sqrt{3}}{2} = s i n 60 °$

Q 21 :

$I f c o t θ = \frac{1}{\sqrt{3}}, t h e v a l u e o f {s e c}^{2} θ + {c o s e c}^{2} θ i s$

1
40/9
38/9
$5 \frac{1}{3}$

1

2

3

4

(4)

Given,

$c o t θ = \frac{1}{\sqrt{3}} = c o t 60 ° \Rightarrow θ = 60 ° N o w, {s e c}^{2} θ + {c o s e c}^{2} θ = {s e c}^{2} 60 ° + {c o s e c}^{2} 60 ° = (2)^{2} + {(\frac{2}{\sqrt{3}})}^{2} = 4 + \frac{4}{3} = \frac{12 + 4}{3} = \frac{16}{3} = 5 \frac{1}{3}$

Q 22 :

Given that $s e c θ = \sqrt{2}, t h e v a l u e o f \frac{1 + t a n θ}{\sin θ} i s$

$2 \sqrt{2}$
$\sqrt{2}$
$3 \sqrt{2}$
2

1

2

3

4

(1)

Given,

$s e c θ = \sqrt{2} \Rightarrow θ = 45 ° N o w, \frac{1 + t a n θ}{\sin θ} = \frac{1 + t a n 45 °}{\sin 45 °} = \frac{1 + 1}{\frac{1}{\sqrt{2}}} = \frac{2}{\frac{1}{\sqrt{2}}} = 2 \sqrt{2}$

Q 23 :

$I f x t a n 60 ° c o s 60 ° = s i n 60 ° c o t 60 °, t h e n x =$

$\cos 30 °$
$\tan 30 °$
$\sin 30 °$
$\cot 30 °$

1

2

3

4

(2)

Given expression is

$x \tan 60 ° c o s 60 ° = s i n 60 ° c o t 60 ° \Rightarrow x \times \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{3}} \Rightarrow x \times \frac{\sqrt{3}}{2} = \frac{1}{2} \Rightarrow x = \frac{1}{\sqrt{3}} = t a n 30 °$

Q 24 :

$I f Δ A B C$ is right angled at C, then the value of $s e c (A + B) i s$

0
1
$\frac{2}{\sqrt{3}}$
not defined

1

2

3

4

(4)

$I n Δ A B C, ∠ C = 90 ° (given) A n d, ∠ A + ∠ B + ∠ C = 180 ° \Rightarrow ∠ A + ∠ B = 90 ° o r A + B = 90 ° \Rightarrow s e c (A + B) = s e c 90 ° = not defined$

Q 25 :

If $I f t a n α + c o t α = 2, t h e n {t a n}^{20} α + {c o t}^{20} α i s e q u a l t o$

0
2
20
$2^{20}$

1

2

3

4

(2)

We have,

$t a n α + c o t α = 2 \Rightarrow t a n α + \frac{1}{\tan α} = 2 \Rightarrow {t a n}^{2} α - 2 t a n α + 1 = 0 \Rightarrow (t a n α - 1)^{2} = 0 \Rightarrow t a n α = 1 \Rightarrow α = 45 ° S o, t a n α = c o t α = 1 N o w, {t a n}^{20} α + {c o t}^{20} α = 1^{20} + 1^{20} = 1 + 1 = 2$

Q 26 :

The value of which of the following is definitely zero?

$(i) s i n 45 ° - c o s 45 ° (i i) t a n 90 ° - c o t 90 ° (i i i) s i n 60 ° - c o s 30 ° (i v) t a n 30 ° - c o t 30 °$

both (i) and (iv)
both (i) and (iii)
only (i)
only (iii)

1

2

3

4

(2)

$(i) s i n 45 ° = \frac{1}{\sqrt{2}}, c o s 45 ° = \frac{1}{\sqrt{2}} \Rightarrow s i n 45 ° - c o s 45 ° = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = 0 (i i) t a n 90 ° = \infty, c o t 90 ° = 0 \Rightarrow t a n 90 ° - c o t 90 ° = not defined (iii) \sin 60 ° = \frac{\sqrt{3}}{2}, \cos 30 ° = \frac{\sqrt{3}}{2} \Rightarrow \sin 60 ° - \cos 30 ° = \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = 0 (iv) \tan 30 ° = \frac{1}{\sqrt{3}}, \cot 30 ° = \sqrt{3} \tan 30 ° - \cot 30 ° = \frac{1}{\sqrt{3}} - \sqrt{3} = \frac{1 - 3}{\sqrt{3}} = - \frac{2}{\sqrt{3}} \neq 0$

Hence, (i) and (iii) will definitely give zero.

Q 27 :

P and Q are acute angles such that P > Q. Then which of the following is definitely true?

$(i) s i n P < s i n Q (i i) t a n P > t a n Q (i i i) c o s P > c o s Q (i v) c o s P > s i n Q$

only (i)
only (ii)
(i) and (ii) both
only (iv)

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2

3

4

(2)

Given that P and Q are acute angles such that P > Q

$L e t P = 60^{°} Q = 30 ° . C h e c k i n g (i) : W e k n o w t h a t \sin θ i n c r e a s e s a s θ i n c r e a s e s f r o m 0 ° t o 90 ° s i n 60 ° = \frac{\sqrt{3}}{2}, s i n 30 ° = \frac{1}{2} \Rightarrow s i n Q < s i n P T h u s (i) i s n o t t r u e C h e c k i n g (i i) : W e k n o w t h a t \tan θ i n c r e a s e s a s θ i n c r e a s e s f r o m 0 ° t o 90 ° \Rightarrow t a n P > t a n Q H e n c e (i i) i s t r u e . C h e c k i n g (i i i) : W e k n o w t h a t \cos θ i n c r e a s e s a s θ i n c r e a s e s f r o m 0 ° t o 90 ° S o (i i i) i s n o t t r u e . C h e c k i n g (i v) : c o s P = c o s 60 ° = \frac{1}{2}, s i n Q = s i n 30 ° = \frac{1}{2} \Rightarrow c o s P = s i n Q$

Topic Question Set

Q 1 :

In $∆ A B C$ right angled at B, $\sin A = \frac{7}{25},$ then the value of cos C is ............................

Q 2 :

If 5 tan θ = 4, then the value of $\frac{5 \sin θ - 3 \cos θ}{5 \sin θ + 2 \cos θ}$ is

Q 3 :

Given that sin α = 1/2 and cos β = 1/2, then the value of (β – α) is

Q 4 :

If sinθ + cosθ = $\sqrt{2}$ , then tanθ + cot θ =

Q 5 :

If 5 tanβ = 4, then $\frac{5 \sin β - 2 \cos β}{5 \sin β + 2 \cos β} =$

Q 6 :

If x tan 60°cos 60°= sin60°cot 60°, then x =

Q 7 :

If sec θ – tan θ = m, then the value of sec θ + tan θ is :

Q 8 :

If cos $(α + β)$ = 0, then value of cos $(\frac{α + β}{2})$ is equal to:

Q 9 :

The value of $(\sin^{2} θ + \frac{1}{1 + \tan^{2} θ})$ is:

Q 10 :

If $\tan^{2} θ + \cot^{2} α = 2,$ where $θ = 45 ° and θ ° \leq α \leq 90 °,$ then the value of a is :

Q 11 :

If $\sin θ = \cos θ,$ $(0 ° < θ < 90 °),$ then value of $(\sec θ . \sin θ)$ is:

Q 12 :

If $5 t a n θ - 12 = 0$ , then the value of $\sin θ$ is

Q 13 :

In $∆ A B C$ , right angled at B $s i n A = \frac{7}{25}$ , then the value of cos C is

Q 14 :

In the given figure, D is the mid-point of BC , then the value of $\frac{\cot y}{\cot x} i s$

Q 15 :

If $a t a n θ = b$ the value of $\frac{b \sin θ - a \cos θ}{b \sin θ + a \cos θ}$ is

Q 16 :

Which of the following trigonometric ratios are correctly written in context of $∆ A B C a s g i v e n ?$

$(i) s i n A = \frac{B C}{A C} (i i) t a n C = \frac{B C}{A B} (i i i) c o s C = \frac{A B}{A C} (i v) s e c A = \frac{A C}{A B}$

Q 17 :

If $s i n α = \frac{\sqrt{3}}{2}, c o s β = \frac{\sqrt{3}}{2}, t h e n t a n α \cdot t a n β i s :$

Q 18 :

The value of $θ$ for which $2 {s i n}^{2} θ = \frac{1}{2}, 0 ° \leq θ \leq 90 ° i s$

Q 19 :

$[\frac{5}{8} {s e c}^{2} 60 ° - {t a n}^{2} 60 ° + {c o s}^{2} 45 °]$ is equal to

Q 20 :

$\frac{2 t a n 30 °}{1 + {t a n}^{2} 30 °}$ is equal to

Q 21 :

$I f c o t θ = \frac{1}{\sqrt{3}}, t h e v a l u e o f {s e c}^{2} θ + {c o s e c}^{2} θ i s$

Q 22 :

Given that $s e c θ = \sqrt{2}, t h e v a l u e o f \frac{1 + t a n θ}{\sin θ} i s$

Q 23 :

$I f x t a n 60 ° c o s 60 ° = s i n 60 ° c o t 60 °, t h e n x =$

Q 24 :

$I f Δ A B C$ is right angled at C, then the value of $s e c (A + B) i s$

Q 25 :

If $I f t a n α + c o t α = 2, t h e n {t a n}^{20} α + {c o t}^{20} α i s e q u a l t o$

Q 26 :

The value of which of the following is definitely zero?

$(i) s i n 45 ° - c o s 45 ° (i i) t a n 90 ° - c o t 90 ° (i i i) s i n 60 ° - c o s 30 ° (i v) t a n 30 ° - c o t 30 °$

Q 27 :

P and Q are acute angles such that P > Q. Then which of the following is definitely true?

$(i) s i n P < s i n Q (i i) t a n P > t a n Q (i i i) c o s P > c o s Q (i v) c o s P > s i n Q$

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Topic Question Set

Q 1 : In ∆ABC right angled at B, sinA=725, then the value of cos C is ............................

Q 2 : If 5 tan θ = 4, then the value of 5sinθ-3cosθ5sinθ+2cosθ is

Q 3 : Given that sin α = 1/2 and cos β = 1/2, then the value of (β – α) is

Q 4 : If sinθ + cosθ = 2, then tanθ + cot θ =

Q 5 : If 5 tanβ = 4, then 5sinβ-2cosβ5sinβ+2cosβ=

Q 6 : If x tan 60°cos 60°= sin60°cot 60°, then x =

Q 7 : If sec θ – tan θ = m, then the value of sec θ + tan θ is :

Q 8 : If cos (α+β) = 0, then value of cos (α+β2) is equal to:

Q 9 : The value of (sin2θ+11+tan2θ) is:

Q 10 : If tan2θ+cot2α=2, where θ=45° and θ°≤α≤90°, then the value of a is :

Q 11 : If sinθ=cosθ, (0°<θ<90°), then value of (secθ.sinθ) is:

Q 12 : If 5 tan θ-12=0, then the value of sin θ is

Q 13 : In ∆ABC, right angled at B sinA=725, then the value of cos C is

Q 14 : In the given figure, D is the mid-point of BC , then the value of cotycotx is

Q 15 : If a tanθ=b the value of bsinθ-acosθbsinθ+acosθ is

Q 16 : Which of the following trigonometric ratios are correctly written in context of ∆ABC as given? (i) sinA=BCAC(ii) tanC=BCAB(iii) cosC=ABAC(iv) secA=ACAB

Q 17 : If sinα=32,cosβ=32, then tanα·tanβ is:

Q 18 : The value of θ for which 2sin 2θ=12,0°≤θ≤90° is

Q 19 : 58sec 260°-tan 260°+cos 245° is equal to

Q 20 : 2tan30°1+tan 230° is equal to

Q 21 : If cotθ=13,the value of sec 2θ+cosec 2θ is

Q 22 : Given that secθ=2, the value of1+tanθsinθ is

Q 23 : If x tan60°cos60°=sin60°cot60°,then x=

Q 24 : If ΔABC is right angled at C, then the value of secA+Bis

Q 25 : If If tanα+cotα=2,then tan 20α+cot 20 α is equal to

Q 26 : The value of which of the following is definitely zero? (i) sin 45°-cos45°(ii) tan 90°-cot90°(iii) sin 60°-cos30°(iv) tan30°-cot30°

Q 27 : P and Q are acute angles such that P > Q. Then which of the following is definitely true? (i) sinP<sinQ(ii) tanP>tanQ(iii) cosP>cosQ (iv) cosP>sinQ

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Q 1 :

In $∆ A B C$ right angled at B, $\sin A = \frac{7}{25},$ then the value of cos C is ............................

Q 2 :

If 5 tan θ = 4, then the value of $\frac{5 \sin θ - 3 \cos θ}{5 \sin θ + 2 \cos θ}$ is

Q 3 :

Given that sin α = 1/2 and cos β = 1/2, then the value of (β – α) is

Q 4 :

If sinθ + cosθ = $\sqrt{2}$ , then tanθ + cot θ =

Q 5 :

If 5 tanβ = 4, then $\frac{5 \sin β - 2 \cos β}{5 \sin β + 2 \cos β} =$

Q 6 :

If x tan 60°cos 60°= sin60°cot 60°, then x =

Q 7 :

If sec θ – tan θ = m, then the value of sec θ + tan θ is :

Q 8 :

If cos $(α + β)$ = 0, then value of cos $(\frac{α + β}{2})$ is equal to:

Q 9 :

The value of $(\sin^{2} θ + \frac{1}{1 + \tan^{2} θ})$ is:

Q 10 :

If $\tan^{2} θ + \cot^{2} α = 2,$ where $θ = 45 ° and θ ° \leq α \leq 90 °,$ then the value of a is :

Q 11 :

If $\sin θ = \cos θ,$ $(0 ° < θ < 90 °),$ then value of $(\sec θ . \sin θ)$ is:

Q 12 :

If $5 t a n θ - 12 = 0$ , then the value of $\sin θ$ is

Q 13 :

In $∆ A B C$ , right angled at B $s i n A = \frac{7}{25}$ , then the value of cos C is

Q 14 :

In the given figure, D is the mid-point of BC , then the value of $\frac{\cot y}{\cot x} i s$

Q 15 :

If $a t a n θ = b$ the value of $\frac{b \sin θ - a \cos θ}{b \sin θ + a \cos θ}$ is

Q 16 :

Which of the following trigonometric ratios are correctly written in context of $∆ A B C a s g i v e n ?$

$(i) s i n A = \frac{B C}{A C} (i i) t a n C = \frac{B C}{A B} (i i i) c o s C = \frac{A B}{A C} (i v) s e c A = \frac{A C}{A B}$

Q 17 :

If $s i n α = \frac{\sqrt{3}}{2}, c o s β = \frac{\sqrt{3}}{2}, t h e n t a n α \cdot t a n β i s :$

Q 18 :

The value of $θ$ for which $2 {s i n}^{2} θ = \frac{1}{2}, 0 ° \leq θ \leq 90 ° i s$

Q 19 :

$[\frac{5}{8} {s e c}^{2} 60 ° - {t a n}^{2} 60 ° + {c o s}^{2} 45 °]$ is equal to

Q 20 :

$\frac{2 t a n 30 °}{1 + {t a n}^{2} 30 °}$ is equal to

Q 21 :

$I f c o t θ = \frac{1}{\sqrt{3}}, t h e v a l u e o f {s e c}^{2} θ + {c o s e c}^{2} θ i s$

Q 22 :

Given that $s e c θ = \sqrt{2}, t h e v a l u e o f \frac{1 + t a n θ}{\sin θ} i s$

Q 23 :

$I f x t a n 60 ° c o s 60 ° = s i n 60 ° c o t 60 °, t h e n x =$

Q 24 :

$I f Δ A B C$ is right angled at C, then the value of $s e c (A + B) i s$

Q 25 :

If $I f t a n α + c o t α = 2, t h e n {t a n}^{20} α + {c o t}^{20} α i s e q u a l t o$

Q 26 :

The value of which of the following is definitely zero?

$(i) s i n 45 ° - c o s 45 ° (i i) t a n 90 ° - c o t 90 ° (i i i) s i n 60 ° - c o s 30 ° (i v) t a n 30 ° - c o t 30 °$

Q 27 :

P and Q are acute angles such that P > Q. Then which of the following is definitely true?

$(i) s i n P < s i n Q (i i) t a n P > t a n Q (i i i) c o s P > c o s Q (i v) c o s P > s i n Q$