Correct Answer – Option 4 : \(\rm 1\over 5\)

Concept:

Inverse trigonometric identities

• sin-1(sin a) = a
• cos-1(cos a) = a
• tan-1(tan a) = a
• cot-1(cot a) = a
• cot^-1 a = 90 – tan^-1 a
• cos-1 a = 90 – sin-1 a

Trigonometric Identities

• cos θ = sin (90 – θ)
• cot θ = tan (90 – θ)

Calculation:

\(\rm \sin\left(\tan^{-1}X\right) = \cos\left(\cot^{-1}{1\over5}\right)\)

⇒ \(\rm \sin\left(\tan^{-1}X\right) = \sin\left[90-\left(\cot^{-1}{1\over5}\right)\right]\)

⇒ \(\rm \tan^{-1}X = 90-\cot^{-1}{1\over5}\)

⇒ \(\rm \tan^{-1}X = 90-\left(90-\tan^{-1}{1\over5}\right)\)

⇒ \(\rm \tan^{-1}X = \tan^{-1}{1\over5}\)

Taking tan both sides

⇒ \(\rm \tan (\tan^{-1}X) = \tan(\tan^{-1}{1\over5})\)

⇒ X = \(\boldsymbol{\rm 1\over5}\)