Correct Answer - Option 4 :
\(\rm 3\)
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:
S = \(\rm 3\tan\left[\cot^{-1}\left\{2\sin\left(\cos^{-1}{\sqrt3\over2}\right)\right\}\right]\)
⇒ S = \(\rm 3\tan\left[\cot^{-1}\left\{2\sin\left(\cos^{-1}{\cos30}\right)\right\}\right]\) (∵ cos 30 = \(\rm \sqrt 3\over2\))
⇒ S = \(\rm 3\tan\left[\cot^{-1}\left\{2\sin\left(30\right)\right\}\right]\)
⇒ S = \(\rm 3\tan\left[\cot^{-1}\left\{2\times{1\over2}\right\}\right]\) (∵ sin 30 = \(\rm 1\over2\))
⇒ S = \(\rm 3\tan\left[\cot^{-1}\left\{1\right\}\right]\)
⇒ S = \(\rm 3\tan\left[\cot^{-1}\left\{\cot45\right\}\right]\)(∵ cot 45 = 1)
⇒ S = \(\rm 3\tan\left[45\right]\)
⇒ S = 3 (∵ tan 45 = 1)
