1.

Prove that(i) cos2 π/4 – sin2 π/12 = √3/4(ii) sin2 (n + 1) A – sin2 nA = sin(2n + 1) A sin A

Answer»

(i) Let us consider the LHS

cos2 π/4 – sin2 π/12

As we know that, cos2A – sinB = cos (A + B) cos (A – B)

Therefore,

cos2 π/4 – sin2 π/12 = cos(π/4 + π/12) cos(π/4 – π/12)

= cos 4π/12 cos 2π/12

= cos π/3 cos π/6

= 1/2 × √3/2

= √3/4

= RHS

∴ LHS = RHS

Thus proved.

(ii) Let us consider the LHS

sin2(n + 1) A – sin2nA

As we know that, sin2A – sinB = sin(A + B) sin(A – B)

Here, A = (n + 1)A and B = nA

Therefore,

sin2(n + 1) A – sin2n A = sin((n + 1) A + nA) sin((n + 1) A – nA)

= sin(nA + A + nA) sin(nA + A – nA)

= sin(2nA + A) sin (A)

= sin(2n + 1) A sin A

= RHS

∴ LHS = RHS

Thus proved.



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