If cos 4x = 1 + k sin²x cos²x, Then Find k
Question:
\[ \cos 4x=1+k\sin^2x\cos^2x \]Find the value of \(k\).
Solution
Using the identity
\[ \cos 4x = 1-2\sin^2 2x \]Also,
\[ \sin 2x=2\sin x\cos x \] \[ \sin^2 2x = 4\sin^2 x\cos^2 x \]Substituting,
\[ \cos 4x = 1-2(4\sin^2 x\cos^2 x) \] \[ \cos 4x = 1-8\sin^2 x\cos^2 x \]Comparing with
\[ \cos 4x = 1+k\sin^2 x\cos^2 x \]we get
\[ k=-8 \]