Question

Consider the following statements and choose the correct identity:

I. sec²θ(1 + sinθ) (1 - sinθ) = -1

II. cot²θ – 1/(1 - cos²θ) = 1

1. Only 1
2. Only 2
3. Both 1 and 2
4. Neither 1 nor 2

Answer 1

Correct Answer - Option 4 : Neither 1 nor 2

Given:

Two identities are given

Formula Used:

Basic concept of trigonometric ratio and identities

We know that

(a² – b²) = (a + b) × (a - b)

sin²θ + cos²θ = 1

cosec²θ - cot²θ = 1

1 + tan²θ = sec²θ

sinθ = 1/cosecθ

Calculation:

The given identity is sec²θ(1 + sinθ) (1 - sinθ) = -1

L.H.S.

∴ Sec²θ(1 + sinθ) (1 - sinθ) = sec²θ (1 - cos²θ) = sec²θ sin²θ

So, statement 1 is incorrect

Now, the given identity is (1 + tan²θ)/ cosec²θ = tan²θ

L.H.S.

∴ cot²θ – 1/(1 - cos²θ) = cot²θ – 1/ sin²θ

⇒ cot²θ – cosec²θ = -1

So, statement 2 is incorrect

Hence, option (4) is correct