The value of $\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+\tan ^{-1} \frac{2}{3}\right)$ is

JEE Main · 2008 · 27 Apr 2008 · Offline · Inverse Trigonometric Functions

The value of

cot (cosec^{- 1} \frac{5}{3} + tan^{- 1} \frac{2}{3})

Answer: (A)

Let

α = cosec^{- 1} \frac{5}{3}

, so

sin α = \frac{3}{5}

cos α = \frac{4}{5}

and

tan α = \frac{3}{4}

, meaning

α = tan^{- 1} \frac{3}{4}

. The required angle is then

tan^{- 1} \frac{3}{4} + tan^{- 1} \frac{2}{3}

, whose tangent is

\frac{\frac{3}{4} + \frac{2}{3}}{1 - \frac{3}{4} \cdot \frac{2}{3}} = \frac{17/12}{1/2} = \frac{17}{6}

. The cotangent of this angle is the reciprocal,

\frac{6}{17}

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