Prove that for real $\theta$, \[\cos(5\theta) = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \thet

Question

Prove that for real $\theta$, $$\cos(5\theta) = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$$ and $$\sin(5\theta) = 5 \cos^4 \theta \sin \theta - 10 \cos^2 \theta \sin^3 \theta + \sin^5 \theta.$$ Hence express $\tan (5 \theta)$ in terms of $\tan \theta$.

Accepted Answer

Both identities follow by real and imaginary parts of $(\cos\theta+i\sin\theta)^5$. $\tan5\theta=\frac{5t-10t^3+t^5}{1-10t^2+5t^4}$, where $t=\tan\theta$ and both tangents exist.

Example for calculating multiple angle formulae

Example for calculating multiple angle formulae

Solution:

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