Assertion (A): For 0<\theta\le{90}^\circ, csc{\theta}-cot{\theta}and csc{\theta}+cot{\theta} are reciprocal of each other. Reason (R): {csc{\ }}^2\theta-{cot{\ }}^2

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Q.
Assertion (A): $F o r 0 < θ \leq 90 °, c s c θ - c o t θ a n d c s c θ + c o t θ$ are reciprocal of each other.

Reason (R): ${c s c}^{2} θ - {c o t}^{2} θ = 1 .$

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)

$S i n c e, {c s c}^{2} θ - {c o t}^{2} θ = 1 \Rightarrow (c s c θ - c o t θ) (c s c θ + c o t θ) = 1 \Rightarrow c s c θ - c o t θ = \frac{1}{c s c θ + c o t θ} o r c s c θ + c o t θ = \frac{1}{c s c θ - c o t θ} (c s c θ - c o t θ) i s r e c i p r o c a l o f (c s c θ + c o t θ) a n d v i c e v e r s a .$

So, both Assertion (A) and Reason (R) are true and R is the correct explanation of A.

Solution

Q.
Assertion (A): $F o r 0 < θ \leq 90 °, c s c θ - c o t θ a n d c s c θ + c o t θ$ are reciprocal of each other.

Reason (R): ${c s c}^{2} θ - {c o t}^{2} θ = 1 .$

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): For 0<θ≤90°,cscθ-cotθ and cscθ+cotθ are reciprocal of each other. Reason (R): csc 2θ-cot 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) Since,csc 2θ-cot 2θ=1⇒cscθ-cotθcscθ+cotθ=1⇒cscθ-cotθ=1cscθ+cotθorcscθ+cotθ=1cscθ-cotθcscθ-cotθ is reciprocal ofcscθ+cotθ and vice versa. So, both Assertion (A) and Reason (R) are true and R is the correct explanation of A.

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Q.
Assertion (A): $F o r 0 < θ \leq 90 °, c s c θ - c o t θ a n d c s c θ + c o t θ$ are reciprocal of each other.

Reason (R): ${c s c}^{2} θ - {c o t}^{2} θ = 1 .$