The value of [2^-135+^-1513] is equal to

Question

The value of $	an\left[2	an^{-1}\dfrac35+\sin^{-1}\dfrac{5}{13}ight]$ is equal to

Accepted Answer

$$2\tan^{-1}\frac35=\tan^{-1}\left(\frac{2\cdot\frac35}{1-\frac{9}{25}}\right)=\tan^{-1}\frac{15}{8}.$$

Let $\theta=\sin^{-1}\dfrac{5}{13}$: $\tan\theta=\dfrac{5}{12}$.

$$\tan\left(\tan^{-1}\frac{15}{8}+\tan^{-1}\frac{5}{12}\right)=\frac{\frac{15}{8}+\frac{5}{12}}{1-\frac{15}{8}\cdot\frac{5}{12}}=\frac{\frac{55}{24}}{\frac{21}{96}}=\frac{55}{24}\cdot\frac{96}{21}=\frac{220}{21}.$$

Solutions · 1