Question

Prove that One minus sin theta into one plus cosec theta is equals to cos theta into cot theta

Answer 1

Step-by-step explanation:

I believe your Question was,

"Prove that

(1 - Sin θ)(1 + Cosec θ) = Cos θ × Cot θ"

We know that,

LHS = (1 - Sin θ)(1 + Cosec θ)

RHS = Cos θ × Cot θ

Now,

(1 - Sin θ)(1 + Cosec θ)

We know that,

Cosec θ = 1/Sin θ

Thus,

(1 - Sin θ)(1 + 1/Sin θ)

Taking LCM, we get,

= (1 - Sin θ)((Sin θ/Sin θ) + (1/Sin θ))

= (1 - Sin θ)((Sinθ + 1)/Sin θ)

= (1 - Sin θ)((1 + Sin θ)/Sin θ)

Using the identity, (a + b)(a - b) = a² - b²

= (1² - Sin²θ)/Sin θ

= (1 - Sin²θ)/Sin θ

Now,

We know that,

Sin²θ + Cos²θ = 1

So,

Cos²θ = 1 - Sin²θ

Thus,

(1² - Sin²θ)/Sin θ

= (Cos²θ)/Sin θ

= (Cos θ × Cos θ)/Sin θ

= Cos θ × (Cos θ/Sin θ)

= Cos θ × Cot θ

LHS = RHS

Hence Proved

Hope it helped and believing you understood it........All the best