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Prove that(i) cos2 π/4 – sin2 π/12 = √3/4(ii) sin2 (n + 1) A – sin2 nA = sin(2n + 1) A sin A |
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Answer» (i) Let us consider the LHS cos2 π/4 – sin2 π/12 As we know that, cos2A – sin2 B = 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 – sin2 B = 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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