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:

$\cos\left(2\sin^{-1}\dfrac35\right)=\cos^2\left(\sin^{-1}\dfrac35\right)-\sin^2\left(\sin^{-1}\dfrac35\right)$

By the Pythagorean theorem, if $\sin x=\dfrac35$, it follows that $\cos x=\dfrac45$:

$\cos^2x+\sin^2x=1\implies \cos x=\pm\sqrt{1-\sin^2x}$

where we take the positive root because $-\dfrac\pi2\le\sin^{-1}x\le\dfrac\pi2$, and $\cos x$ is non-negative throughout this domain.

So we have

$\cos^2\left(\sin^{-1}\dfrac35\right)-\sin^2\left(\sin^{-1}\dfrac35\right)=\left(\dfrac45\right)^2-\left(\dfrac35\right)^2=\dfrac7{25}$