Q) Prove that: (sin A + sec A)² + (cos A + cosec A)² = (1 + sec A cosec A)².

(Q 29 B – 30/3/3 – CBSE 2026 Question Paper)

Ans:

Let’s start from LHS:

∴ LHS = (sin A + sec A) ² + (cos A + cosec A) ²

By algebraic identity, we have: (a + b) ² = a ² + b ² + 2 a b

∴ LHS = (sin A + sec A) ² + (cos A + cosec A) ²

∴ LHS = (sin ² A + sec ² A + 2 sin A sec A) + (cos ² A + cosec ² A + 2 cos A cosec A)

∴ LHS = (sin ² A + + 2 sin A ) + (cos ² A + + 2 cos A )

∴ LHS = sin ² A + + 2 + cos ² A + + 2

∴ LHS = sin ² A + cos ² A + + + 2 + 2

∴ LHS = sin ² A + cos ² A + + 2

By trigonometric identity, we have sin ² θ + cos ² θ = 1

∴ LHS = sin ² A + cos ² A + + 2

∴ LHS = 1 + + 2

∴ LHS = 1 + +

∴ LHS = 1 +

∴ LHS =

If we consider, sin A . cos A = P, then:

∴ LHS =

∴ LHS =          (∵ (a + b) ² = a ² + b ² + 2 a b)

∴ LHS =

Let’s substitute back the value of P = sin A . cos A in LHS:

∴ LHS =

∴ LHS =

∴ LHS =

∴ LHS = (1 + cosec A . sec A) ²

∴ LHS = (1 + sec A . cosec A) ² = RHS

Hence Proved !

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