Question -
Answer -
(i) cos2 π/4 – sin2 π/12 =√3/4
Let us consider LHS:
cos2 π/4 – sin2 π/12
We know that, cos2A – sin2 B= cos (A + B) cos (A – B)
So,
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
Hence proved.
(ii) sin2 (n + 1) A – sin2nA =sin (2n + 1) A sin A
Let us consider LHS:
sin2 (n + 1) A – sin2nA
We know that, sin2A – sin2 B= sin (A + B) sin (A – B)
Here, A = (n + 1) A and B = nA
So,
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
Hence proved.