What is the value of ≤ft(^{-1} 1/3) + (2 ^{-1} 3) ?

Correct Answer is : 10−810{{10} - 8}{10}1010−8

Correct Answer is : 10−810{{10} - 8}{10}1010−8

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UPSC NDA Math Model Paper 8

Question : 82 of 120

Marks: +1, -0

\sin \left(\tan^{-1} \frac{1}{3}\right) + \cos (2 \tan^{-1} 3) ?

Solution:

We know that

\csc^2 \theta = 1 + \cot^2 \theta

\Rightarrow \sin^2 \theta = \frac{1}{\csc^2 \theta} = \frac{1}{1 + \cot^2 \theta}

= \frac{1}{1 + \frac{1}{\tan^2 \theta}} = \frac{\tan^2 \theta}{1 + \tan^2 \theta}

\Rightarrow \sin \theta = \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}} .

Let's say that

\tan^{-1} \frac{1}{3} = x

and

\tan^{-1} 3 = y

\Rightarrow \tan x = \frac{1}{3}

and

\tan y = 3

\therefore

The given expression

\sin \left(\tan^{-1} \frac{1}{3}\right) + \cos (2 \tan^{-1} 3)

can be written as:

\sin (x) + \cos (2y)

= \frac{\tan x}{\sqrt{1 + \tan^2 x}} + \frac{1 - \tan^2 y}{1 + \tan^2 y}

= \frac{\frac{1}{3}}{\sqrt{1 + \left(\frac{1}{3}\right)^2}} + \frac{1 - 3^2}{1 + 3^2}

= \frac{1}{\sqrt{10}} - \frac{8}{10}

= \frac{\sqrt{10} - 8}{10} .

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