Question

Evaluate \[\cot \left( {{{\cos }^{ - 1}}\left( {\dfrac{7}{{25}}} \right)} \right)\].
(A) $\dfrac{{25}}{{24}}$
(B) $\dfrac{{25}}{7}$
(C) $\dfrac{{24}}{{25}}$
(D) None of these

Answer 1

Hint: In the given problem, we are required to calculate cotangent of an angle whose cosine is provided to us in the question itself. For finding a trigonometric ratio for an angle given in terms of an inverse trigonometric ratio, we have to first assume that angle to be some unknown, let's say $\theta $. Then proceeding further, we have to find the value of a trigonometric function of that unknown angle $\theta $. Then we find the required trigonometric ratio with help of basic trigonometric formulae and definitions of trigonometric ratios.

Complete step-by-step solution:
So, in the given problem, we have to find the value of\[\cot \left( {{{\cos }^{ - 1}}\left( {\dfrac{7}{{25}}} \right)} \right)\].
Hence, we have to find the cotangent of the angle whose cosine is given to us as \[\left( {\dfrac{7}{{25}}} \right)\].
Let us assume $\theta $ to be the concerned angle.
Then, $\theta = {\cos ^{ - 1}}\left( {\dfrac{7}{{25}}} \right)$
Taking cosine on both sides of the equation, we get
 $ = \cos \theta = \left( {\dfrac{7}{{25}}} \right)$
To evaluate the value of the required expression, we must keep in mind the formulae of basic trigonometric ratios.
We know that, $\cot \theta = \dfrac{{{\text{Base}}}}{{{\text{Perpendicular}}}}$ and $\cos \theta = \dfrac{{{\text{Base}}}}{{{\text{Hypotenuse}}}}$.
So, $\cos \theta = \dfrac{{{\text{Base}}}}{{{\text{Hypotenuse}}}} = \dfrac{7}{{25}}$
Let length of base be $7x$.
Then, length of hypotenuse $ = 25x$.
Now, applying Pythagoras Theorem,
${\left( {Hypotenuse} \right)^2} = {\left( {Base} \right)^2} + {\left( {Perpendicular} \right)^2}$
$ = {\left( {25x} \right)^2} = {\left( {7x} \right)^2} + {\left( {{\text{Perpendicular}}} \right)^2}$
$ = 625{x^2} = 49{x^2} + {\left( {{\text{Perpendicular}}} \right)^2}$
$ = {\left( {{\text{Perpendicular}}} \right)^2} = 625{x^2} - 49{x^2}$
$ = {\left( {{\text{Perpendicular}}} \right)^2} = 576{x^2}$
Taking square root on both sides of the equation, we get,
$ = \left( {{\text{Perpendicular}}} \right) = \sqrt {576} x$
We know the square root of $576$ is $24$. Hence, we get,
$ = \left( {{\text{Perpendicular}}} \right) = 24x$
So, we get ${\text{Perpendicular}} = 24x$
Hence, $\cot \theta = \dfrac{{{\text{Base}}}}{{{\text{Perpendicular}}}} = \dfrac{7}{{24}}$
So, the value of\[\cot \left( {{{\cos }^{ - 1}}\left( {\dfrac{7}{{25}}} \right)} \right)\] is $\dfrac{7}{{24}}$.
Hence, the correct answer to the problem is option (D) None of these.

Note: Such problems require basic knowledge of trigonometric ratios and formulae. Besides this, knowledge of concepts of inverse trigonometry is extremely essential to answer these questions correctly. We must know the definitions of trigonometric ratios sine, cosine and cotangent to solve the problem. Such questions require clarity of basic concepts of trigonometric functions as well as their inverse.