Question

10.) Find the exact value of cos⁡(2 sin^(−1)⁡(3/5) ). Do not use a calculator and show your work!

Answer 1

Double angle identity for cosine:


`\cos\left(2\sin^(-1)\frac35\right)=\cos^2\left(\sin^(-1)\frac35\right)-\sin^2\left(\sin^(-1)\frac35\right)`

By the Pythagorean theorem, if
`\sin x=\frac35`, it follows that
`\cos x=\frac45`:


`\cos^2x+\sin^2x=1\implies \cos x=\pm√(1-\sin^2x)`

where we take the positive root because
`-\frac\pi2\le\sin^(-1)x\le\frac\pi2`, and
`\cos x` is non-negative throughout this domain.

So we have


`\cos^2\left(\sin^(-1)\frac35\right)-\sin^2\left(\sin^(-1)\frac35\right)=\left(\frac45\right)^2-\left(\frac35\right)^2=\frac7{25}`