22 b. Prove that: 1 + (cot^2 α)/(1 + cosec α) = cosec α

Question

Q) Prove that: 1 + $\frac{(\cot^2 \alpha)}{(1 + \cos ec \alpha)}$ = cosec α

(Q 22 B - 30/2/2 - CBSE 2026 Question Paper)

Sapling Academy · Accepted Answer

Let's start from LHS:

LHS = 1 + $\frac{(\cot^2 \alpha)}{(1 + \cos ec \alpha)}$

Step 1: From Trigonometric identities, we know that

1 + cot 2 α = cosec 2 α

∴ cot 2 α = cosec 2 α - 1

Step 2: Let's substitute the value of cot 2 α from stel 1:

LHS = 1 + $\frac{(\cot^2 \alpha)}{(1 + \cos ec \alpha)}$

= 1 + $\frac{(\cos ec^2 \alpha - 1)}{(1 + \cos ec \alpha)}$

= 1 + $\frac{(\cos ec \alpha - 1)(\cos ec \alpha + 1)}{(1 + \cos ec \alpha)}$

= 1 + $\frac{(\cos ec \alpha - 1)\cancel{(\cos ec \alpha + 1)}}{\cancel{(\cos ec \alpha + 1)}}$

= 1 + (cosec α - 1)

= cosec α = RHS

Hence Proved !

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