Prove that cos(2x)=2cos²x-1
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Solution
\[\begin{aligned}
\cos (2 x) &=\frac{e^{2 i x}+e^{-2 i x}}{2}=\frac{\left(e^{i x}\right)^{2}+\left(e^{-i x}\right)^{2}}{2} \\\\
&=\frac{(\cos x+i \sin x)^{2}+(\cos x-i \sin x)^{2}}{2} \\\\
&=\frac{\cos ^{2} x+2 i \cos x \sin x-\sin ^{2} x+\cos ^{2} x-2 i \cos x \sin x-\sin ^{2} x}{2} \\\\
&=\frac{\cos ^{2} x-\sin ^{2} x+\cos ^{2} x-\sin ^{2} x}{2}=\frac{2 \cos ^{2} x-2 \sin ^{2} x}{2} \\\\
&=\cos ^{2} x-\sin ^{2} x=\cos ^{2} x-\left(1-\cos ^{2} x\right) \\\\
&=\cos ^{2} x-1+\cos ^{2} x \\\\
&=2 \cos ^{2} x-1
\end{aligned}\]
Home -> Solved problems -> Prove that cos(2x)=2cos²x-1
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