Trignometric problem (using De Movier's Theorem) !enter image description here **Ok so this question, I started out writing tan as sin and cos in the right side of the equation, simplified as much as possible and ended up with a very (sort of) fascinating equation which is** !enter image description here _Where s = sin thetha and c= sin thetha_ As you can see the denominator and numerator looks very simple where on top sin x is with the power 5 and in the bottom cos x is with the power 5. **Then I went onto applying the De Movire theorem** And then ended up with- !enter image description here As you can see the required terms sin 5 thetha and cos 5 thetha is there, but now all I need is to cancel out those two other terms in both denominator and numerator. I simply cant find a way in which those cancel off, My question is- **How do I proceed from here?? And if I had done something wrong (if those dont cancel) then where did I go wrong? Please help**

Using DeMoivre's Theorem and the Binomial Theorem, you get:

$\cos 5\theta + i\sin 5\theta = (\cos \theta + i\sin \theta)^5$

$= \cos^5\theta + 5i\cos^4\theta\sin\theta + 10i^2\cos^3 \theta\sin^2\theta + 10i^3\cos^2 \theta\sin^3\theta + 5i^4\cos \theta \sin^4 \theta + i^5\sin^5\theta$

$= \cos^5\theta + 5i\cos^4\theta\sin \theta - 10\cos^3 \theta\sin^2\theta - 10i\cos^2 \theta\sin^3\theta + 5\cos\theta\sin^4 \theta + i\sin^5\theta$

$= (\cos^5\theta - 10\cos^3\theta\sin^2\theta + 5\cos\theta\sin^4 \theta)+i(5\cos^4\theta\sin \theta - 10\cos^2 \theta\sin^3\theta + \sin^5\theta)$

Equate real and imaginary parts to get expressions for $\cos 5\theta$ and $\sin 5\theta$. Can you finish from here?