Assertion (A): {sec{\ }}^2\theta-{tan{\ }}^2\theta=1\ \mathrm{for\ all\ values\ of\ }\theta\neq{90}^\circ. Reason (R): {sin{\ }}^2\theta+{cos{\ }}^2

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Q.
$A s s e r t i o n (A) : {s e c}^{2} θ - {t a n}^{2} θ = 1 for all values of θ \neq 90 ° .$

$R e a s o n (R) : \sin^{2} θ + \cos^{2} θ = 1$

1 Both A and R are true, and R is the correct explanation of A.

2 Both A and R are true, but R is not the correct explanation of A.

3 A is true, but R is false

4 A is false, but R is true

Ans.
(1)

Both Assertion and Reason are true, and the Reason is the correct explanation of the Assertion. If you divide the basic Pythagorean identity

${s i n}^{2} θ + {c o s}^{2} θ = 1 b y {c o s}^{2} θ$ then you get $1 + {t a n}^{2} θ = {s e c}^{2} θ \Rightarrow {s e c}^{2} θ - {t a n}^{2} θ = 1 .$

Solution

Q.
$A s s e r t i o n (A) : {s e c}^{2} θ - {t a n}^{2} θ = 1 for all values of θ \neq 90 ° .$

$R e a s o n (R) : \sin^{2} θ + \cos^{2} θ = 1$

1 Both A and R are true, and R is the correct explanation of A.

2 Both A and R are true, but R is not the correct explanation of A.

3 A is true, but R is false

4 A is false, but R is true

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Solution

Q. Assertion(A): sec 2θ-tan 2θ=1 for all values of θ≠90°. Reason(R): sin 2θ+cos 2θ=1

1 Both A and R are true, and R is the correct explanation of A.

2 Both A and R are true, but R is not the correct explanation of A.

3 A is true, but R is false

4 A is false, but R is true

Ans. (1) Both Assertion and Reason are true, and the Reason is the correct explanation of the Assertion. If you divide the basic Pythagorean identity sin 2θ+cos 2θ=1 by cos 2θ then you get 1+tan 2θ=sec 2θ⇒sec 2θ-tan 2θ=1.

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Q.
$A s s e r t i o n (A) : {s e c}^{2} θ - {t a n}^{2} θ = 1 for all values of θ \neq 90 ° .$

$R e a s o n (R) : \sin^{2} θ + \cos^{2} θ = 1$