(^-135+^-1513+^-13365) is equal to:

Question

$\cos\left(\sin^{-1}\dfrac35+\sin^{-1}\dfrac{5}{13}+\sin^{-1}\dfrac{33}{65}\right)$ is equal to:

Accepted Answer

Convert to tangents: $\sin^{-1}\dfrac35=\tan^{-1}\dfrac34$, $\sin^{-1}\dfrac{5}{13}=\tan^{-1}\dfrac{5}{12}$, $\sin^{-1}\dfrac{33}{65}=\tan^{-1}\dfrac{33}{56}$.

$$\tan^{-1}\frac34+\tan^{-1}\frac{5}{12}=\tan^{-1}\left(\frac{\frac34+\frac{5}{12}}{1-\frac34\cdot\frac{5}{12}}\right)=\tan^{-1}\frac{56}{33}.$$

Then
$$\tan^{-1}\frac{56}{33}+\tan^{-1}\frac{33}{56}=\tan^{-1}\frac{56}{33}+\cot^{-1}\frac{56}{33}=\frac{\pi}{2}.$$

$$\cos\left(\frac{\pi}{2}\right)=0.$$

Solutions · 1