Question

Find the value of $\cos \left( {{{\tan }^{ - 1}}\tan 4} \right)$.
A) $\dfrac{1}{{\sqrt {17} }}$
B) $- \dfrac{1}{{\sqrt {17} }}$
C) $\cos 4$
D) $- \cos 4$

Answer 1

Use the property of $\tan \theta$ to convert the radian angle, $\tan 4$ into degrees and also use allied angles to write the angle within the range of $\tan \theta$. Once the angle is converted use inverse trigonometric property for ${\tan ^{ - 1}}\left( {\tan \theta } \right)$ and simplify.

Complete step by step solution:
We know that the property of tan inverse of tan function is ${\tan ^{ - 1}}\left( {\tan \theta } \right) = \theta$ , where $\theta$ should lie in the interval $\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)$.
Since, the value of $4$ is greater than $\dfrac{\pi }{2}$, rewrite the angle $\tan 4$ using the allied angles of trigonometry to restrict the value within the range of $\tan \theta$.
$\Rightarrow \tan 4 = - \tan \left( {\pi - 4} \right)$
As we know that $\tan \left( { - \theta } \right) = - \tan \theta$, then
$\Rightarrow \tan 4 = \tan \left( {4 - \pi } \right)$
Now, As the angle $\tan 4$ is converted into tangent angle less than $\dfrac{\pi }{2}$, substitute $\tan 4 = \tan \left( {4 - \pi } \right)$ in the given expression.
$\Rightarrow \cos \left( {{{\tan }^{ - 1}}\tan 4} \right) = \cos \left( {{{\tan }^{ - 1}}\tan \left( {4 - \pi } \right)} \right)$
Use the property of inverse, then the above expression will be simplified as below.
$\Rightarrow \cos \left( {{{\tan }^{ - 1}}\tan 4} \right) = \cos \left( {4 - \pi } \right)$
Use the condition for cosine function $\cos \left( { - \theta } \right) = \cos \theta$ and rewrite the above cosine angle as,
$\Rightarrow \cos \left( {4 - \pi } \right) = \cos \left[ { - \left( {\pi - 4} \right)} \right]$
$\Rightarrow \cos \left( {4 - \pi } \right) = \cos \left( {\pi - 4} \right)$
So, the expression becomes,
$\Rightarrow \cos \left( {{{\tan }^{ - 1}}\tan 4} \right) = \cos \left( {\pi - 4} \right)$
We know that the allied angle formula for cosine angle of trigonometry is $\cos \left( {\pi - \theta } \right) = - \cos \theta$.
Now, use the above formula to solve the given trigonometric expression.
$\Rightarrow \cos \left( {{{\tan }^{ - 1}}\tan 4} \right) = - \cos 4$

Therefore, the option (D) is correct.

Note:
In these types of questions, first convert the given angle in radian into degrees.
Check if the given angle is within the range of the given trigonometric ratio, if the given radian angle has value in degrees greater than the range, use allied angles to get the angle within the range.
Make sure signs are taken very carefully while converting the angles to some other allied angles, if it's not done properly, it may lead to the incorrect answer.