cosec[2^-1(5)+^-145] is equal to

Question

$	ext{cosec}\left[2\cot^{-1}(5)+\cos^{-1}\dfrac45ight]$ is equal to

Accepted Answer

$2\cot^{-1}5=2\tan^{-1}\dfrac15=\tan^{-1}\left(\dfrac{2/5}{1-1/25}\right)=\tan^{-1}\dfrac{5}{12}$.

Let $x=\tan^{-1}\dfrac{5}{12}$: $\sin x=\dfrac{5}{13},\cos x=\dfrac{12}{13}$. Let $y=\cos^{-1}\dfrac45$: $\sin y=\dfrac35$.

$$\text{cosec}(x+y)=\frac{1}{\sin x\cos y+\cos x\sin y}=\frac{1}{\frac{5}{13}\cdot\frac45+\frac{12}{13}\cdot\frac35}=\frac{1}{\frac{4}{13}+\frac{36}{65}}=\frac{1}{\frac{56}{65}}=\frac{65}{56}.$$

Solutions · 1