Answer
Ans. Cos4$\theta$=$cos^4θ+sin^4θ-6sin^2θcos^2θ$
Or
=$8cos^4θ-8cos^2θ+1$
Work Step by Step
Given:- By De moivre's theorem
$(cosθ+isinθ)^n=(cosnθ+sinnθ) $ ~Eq.1
If n=4
$(cosθ+isinθ)^4$=$(cos^2θ+2isinθcosθ-sin^2θ)(cos^2θ+2isinθcosθ-sin^2θ)$
=$(cos^4θ-6sin^2θcos^2θ+sin^4θ)+i(4sinθcos^3θ-4sin^3θcosθ)$
After comparing of Eq.1 LHS and RHS we get
Cos4$\theta$=$(cos^4θ-6sin^2θcos^2θ+sin^4θ)$
Put $sin^2θ=1-cos^2θ$ in the above equation
=$8cos^4θ-8cos^2θ+1$
