Associate angles | Example-Simplification

Simplification problem example:

Solve:
LHS \(\cos18^{o}+\cos162^{o}+\cos234^{o}+\cos306^{o}\)
=\(\cos18^{o}+\cos(90^{o}\times2-18^{o})+\cos(90^{o}\times2+54^{o})+\cos(90^{o}\times4-54^{o})\)
=\(\cos18^{o}-\cos18^{o}-\cos54^{o}+\cos54^{o}\)
=0=RHS (Proved)

Solve:
\(\tan\frac{3\pi}{20}\tan\frac{4\pi}{20}\tan\frac{5\pi}{20}\tan\frac{6\pi}{20}\tan\frac{7\pi}{20}\)
=\(\tan\frac{3\pi}{20}\tan\frac{4\pi}{20}\tan\frac{\pi}{4}\tan\Big(\frac{\pi}{2}-\frac{4\pi}{20}\Big)\tan\Big(\frac{\pi}{2}-\frac{3\pi}{20}\Big)\)
=\(\tan\frac{3\pi}{20}\tan\frac{4\pi}{20}.1.\cot\frac{4\pi}{20}\frac{3\pi}{20}\)
=1 Ans

Solve:
LHS=\(\cos^{2}\frac{\pi}{4}+\sin^{2}\frac{3\pi}{4}+\sin^{2}\frac{5\pi}{4}+\sin^{2}\frac{7\pi}{4}\)
=\(\cos^{2}\frac{\pi}{4}+\sin^{2}(\frac{\pi}{2}+\frac{\pi}{4})+\sin^{2}(\pi+\frac{\pi}{4})+\sin^{2}(2\pi-\frac{\pi}{4})\)
=\(\cos^{2}\frac{\pi}{4}+\cos^{2}\frac{\pi}{4}+\sin^{2}\frac{\pi}{4}+\sin^{2}\frac{\pi}{4}\)
=\(2(\cos^{2}\frac{\pi}{4}+\sin^{2}\frac{\pi}{4})\)
=2 Ans