Question

Prove that : 2$\sin^2 27$ = (cos36-cos54

Answer 1

The identity is false.

Answer 2

The identity $2\sin^2 27^\circ = \cos 36^\circ - \cos 54^\circ$ is evaluated by simplifying both sides. The LHS, $2\sin^2 27^\circ$, simplifies to $1 - \cos 54^\circ$ using the identity $2\sin^2 \theta = 1 - \cos 2\theta$. The RHS remains $\cos 36^\circ - \cos 54^\circ$. Equating the simplified LHS and RHS, we get $1 - \cos 54^\circ = \cos 36^\circ - \cos 54^\circ$. This further simplifies to $1 = \cos 36^\circ$. Since the known value of $\cos 36^\circ$ is $\frac{\sqrt{5}+1}{4}$, which is not equal to 1, the original identity is false.