Q) Prove that (cos A – sin A + 1) / (cos A + sin A – 1) = cosec A + cot A
Q29 B– Sample Question Paper – Set 1 – Maths Standard – CBSE 2026
Ans:
Let’s start from LHS:
LHS = (cos A – sin A + 1) / (cos A + sin A – 1)
To normalize the denominator, let’s multiple numerator and denominator both by (cos A + sin A + 1):
LHS = (cos A – sin A + 1) / (cos A + sin A – 1) x (cos A + sin A + 1) / (cos A + sin A + 1)
= (cos A – sin A + 1) x (cos A + sin A + 1) / (cos A + sin A – 1) (cos A + sin A + 1)
= ((cos A + 1) – sin A) x ((cos A + 1) + sin A) / ((cos A + sin A) – 1) ((cos A + sin A) + 1)
= ((cos A + 1)2 – sin2 A) / (cos A + sin A)2 – 1)
= (cos2 A + 1 + 2 cos A – sin2 A) / (cos2 A + sin2 A + 2 sin A cos A – 1)
Since cos2 A + sin2 A = 1
∴ LHS = (cos2 A + (cos2 A + sin2 A ) + 2 cos A – sin2 A) / (1 + 2 sin A cos A – 1)
= (cos2 A + cos2 A + sin2 A + 2 cos A – sin2 A) / (2 sin A cos A )
= (2 cos2 A + 2 cos A) / (2 sin A cos A )
= (cos2 A + cos A) / (sin A cos A )
= cos2 A / sin A cos A + cos A / sin A cos A
= cos A / sin A + 1 / sin A
= cot A + cosec A = RHS
Hence Proved!
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