Solveeit Logo
Login

Question

Question: The value of \(\sin\left( 2\tan^{- 1}\left( \frac{1}{3} \right) \right) + \cos(\tan^{- 1}2\sqrt{2}) ...

The value of sin⁡(2tan⁡−1(13))+cos⁡(tan⁡−122)=\sin\left( 2\tan^{- 1}\left( \frac{1}{3} \right) \right) + \cos(\tan^{- 1}2\sqrt{2}) =

A

1615\frac{16}{15}

B

1415\frac{14}{15}

C

1215\frac{12}{15}

D

1115\frac{11}{15}

Answer

1415\frac{14}{15}

Explanation

Solution

sin⁡[2tan⁡−1(13)]+cos⁡[tan⁡−1(22)]\mathbf{\sin}\left\lbrack \mathbf{2}\mathbf{\tan}^{\mathbf{- 1}}\left( \frac{\mathbf{1}}{\mathbf{3}} \right) \right\rbrack\mathbf{+}\mathbf{\cos}\mathbf{\lbrack}\mathbf{\tan}^{\mathbf{- 1}}\mathbf{(}\mathbf{2}\sqrt{\mathbf{2}}\mathbf{)\rbrack} = sin⁡[tan⁡−1231−19]+cos⁡[tan⁡−1(22)]\sin\left\lbrack \tan^{- 1}\frac{\frac{2}{3}}{1 - \frac{1}{9}} \right\rbrack + \cos\lbrack\tan^{- 1}(2\sqrt{2})\rbrack

=sin⁡[tan⁡−134]+cos⁡[tan⁡−122]= \sin\left\lbrack \tan^{- 1}\frac{3}{4} \right\rbrack + \cos\lbrack\tan^{- 1}2\sqrt{2}\rbrack

= sin⁡[sin⁡−135]+cos⁡[cos⁡−113]=35+13=1415\sin\left\lbrack \sin^{- 1}\frac{3}{5} \right\rbrack + \cos\left\lbrack \cos^{- 1}\frac{1}{3} \right\rbrack = \frac{3}{5} + \frac{1}{3} = \frac{14}{15}