Evaluate

OR

If cosec² θ (1 + cos θ) (1 – cos θ) = k, then find the value of k.

Given:

[∵, tan θ = cot (90° - θ)]

= 1

OR

Given: cosec² θ (1 + cos θ) (1 – cos θ) = k

To find: Value of k

Proof: Taking LHS

= cosec² θ (1 + cos θ) (1 – cos θ)

Using the identity,

(a + b)(a – b) = (a² – b²)

= cosec² θ [(1)² – (cosθ)²]

= cosec² θ [ 1 – cos²θ] …(i)

We know that,

cos²θ + sin²θ = 1

or sin²θ = 1 – cos²θ

So, equation (i) becomes,

= cosec²θ (sin²θ) …(ii)

Now, we also know that,

So, equation (ii) becomes

= 1

∴ cosec² θ (1 + cos θ) (1 – cos θ) = 1 = k

Hence, the value of k is 1