Question

If $\cos x = \tan {40^ \circ }\tan {50^ \circ }$, then what is the value of $x$?

Answer 1

Hint: The given question deals with basic simplification of trigonometric functions by using some of the simple complementary trigonometric formulae such as $\tan x = \cot \left( {{{90}^ \circ } - x} \right)$ . Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem. Evaluating the right side of the equation, we have to solve a trigonometric equation.

Complete step by step solution:
In the given problem, we have to find the value of x for which the trigonometric equation $\cos x = \tan {40^ \circ }\tan {50^ \circ }$ holds true. Now, we have to simplify the product of the trigonometric functions given to us in the right side of the equation.
So, we have,
$\cos x = \tan {40^ \circ }\tan {50^ \circ }$
Now, we know that tangent and cotangent are complementary ratios. So, $\tan x = \cot \left( {{{90}^ \circ } - x} \right)$.
Hence, we get,
\[ \Rightarrow \cos x = \tan \left( {{{40}^ \circ }} \right)\cot \left( {{{90}^ \circ } - {{50}^ \circ }} \right)\]
Simplifying further, we get,
\[ \Rightarrow \cos x = \tan \left( {{{40}^ \circ }} \right)\cot \left( {{{40}^ \circ }} \right)\]
Now, we know that the tangent and cotangent are reciprocal ratios of each other. So, the product of tangent and cotangent for the same angle is one. Hence, we get,
\[ \Rightarrow \cos x = 1\]
So, we have arrived at a trigonometric equation by solving the original trigonometric equation $\cos x = \tan {40^ \circ }\tan {50^ \circ }$.
Now, we have to solve the simplified trigonometric equation \[\cos x = 1\].
So, we know that the value of $\cos 0$ is one. Hence, we get the solution for \[\cos x = 1\] as all the multiples of $2n\pi $, where n is any integer.
Hence, the solution for the value of x in the trigonometric equation $\cos x = \tan {40^ \circ }\tan {50^ \circ }$ is $2n\pi $, where n is any integer.

Note:
Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart. Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such types of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations. However, questions involving this type of simplification of trigonometric ratios may also have multiple interconvertible answers.