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Mathematics

Eleven Standard >> Associate angles | Example-Simplification

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Simplification problem example:

1. Show that \(\cos18^{o}+\cos162^{o}+\cos234^{o}+\cos306^{o}=0\)

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)

2. Find the value of \(\tan\frac{3\pi}{20}\tan\frac{4\pi}{20}\tan\frac{5\pi}{20}\tan\frac{6\pi}{20}\tan\frac{7\pi}{20}\)

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

3. Show that \(\cos^{2}\frac{\pi}{4}+\sin^{2}\frac{3\pi}{4}+\sin^{2}\frac{5\pi}{4}+\sin^{2}\frac{7\pi}{4}\)=2

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 
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