Verify that
$$\cos 3 heta = 4\cos heta \cdotp \cos(60\degree - heta )\cos(60\degree + heta ) $$

Question

Verify that 
$$\cos 3	heta  = 4\cos 	heta \cdotp \cos(60\degree -	heta )\cos(60\degree +	heta )  $$

Uniteasy Admin · Accepted Answer

Using the product to sun identity
$$\cos \alpha\cos \beta = \dfrac{1}{2} [\cos(\alpha+\beta ) +\cos(\alpha - \beta) ]$$
$$4\cos 	heta \cdotp \cos(60\degree -	heta )\cos(60\degree +	heta ) $$
$$=2\cos 	heta (\cos 120\degree +\cos 2	heta )$$
$$=2\cos 	heta (-\dfrac{1}{2}+2\cos^2 	heta -1)  $$
$$=-3\cos 	heta +4\cos^3 	heta  $$
$$=\cos 3	heta $$

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