1.

Prove the identity: sin2(π/8 + x/2) – sin2(π/8 – x/2) = 1/√2 sin x

Answer»

Let us consider the LHS

sin2(π/8 + x/2) – sin2(π/8 – x/2)

As we know, sin2 A – sin2 B = sin(A + B) sin(A - B)

Therefore,

sin2(π/8 + x/2) – sin2(π/8 – x/2) = sin (π/8 + x/2 + π/8 – x/2) sin (π/8 + x/2 – (π/8 – x/2))

= sin (π/8 + π/8) sin (π/8 + x/2 – π/8 + x/2)

= sin π/4 sin x

= 1/√2 sin x [since, π/4 = 1/√2]

= RHS

Thus proved.



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