1.

Find X if; \(\rm \cos\left(\cot^{-1}{1\over5}\right) = \sin\left(\tan^{-1}X\right)\)1. \(\rm {5\over 4}\)2. 53. \(\rm {4\over 5}\)4. \(\rm 1\over 5\)

Answer» 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 = \(\boldsymbol{\rm 1\over5}\) 


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