Question

Assertion (A): If k+1=sec²θ(1+sinθ)(1−sinθ), then the value of k is 1.

Reason (R): If sinθ+cosθ=√3, then the value of tanθ+cotθ is 1.

A. A is false but R is true.
B. Both A and R are true and R is not the correct explanation of A.
C. Both A and R are true and R is the correct explanation of A.
D. A is true but R is false.

Answer 1

If k+1=sec²θ(1+sinθ)(1−sinθ), then the value of k is 1. If sinθ+cosθ=√3, then the value of tanθ+cotθ is 1. Both the Assertion and Reason are true. So the correct answer is option C.

Step-by-step explanation:

Assertion (A): If k+1=sec²θ(1+sinθ)(1−sinθ), then the value of k is 1.
Reason (R): If sinθ+cosθ=√3, then the value of tanθ+cotθ is 1.

To determine the validity of the assertion and reason, we need to evaluate both statements separately.

For Assertion (A):

k + 1 = sec²θ(1 + sinθ)(1 − sinθ)

Using trigonometric identities, we can simplify this equation:

k + 1 = (1/cos²θ)(1 − sin²θ) = (1/cos²θ)(cos²θ) = 1

Therefore, Assertion (A) is true.

For Reason (R):

sinθ + cosθ = √3

Squaring both sides of the equation:

sin²θ + 2sinθcosθ + cos²θ = 3

Using the trigonometric identity sin²θ + cos²θ = 1:

1 + 2sinθcosθ = 3

2sinθcosθ = 2

Simplifying further:

sinθcosθ = 1

tanθ = sinθ/cosθ = 1

cotθ = cosθ/sinθ = 1

Therefore, Reason (R) is also true.

Since both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A), the correct answer is C. Both A and R are true and R is the correct explanation of A.