Question

How do you solve the equation $\dfrac{{\tan {{85}^ \circ } - \tan {{25}^ \circ }}}{{1 + \tan {{85}^ \circ }\tan {{25}^ \circ }}}$ and find its value?

Answer 1

For solving these types of questions it is very important to learn the values of trigonometric ratios like cos 30, tan30 etc. and to remember the trigonometric formulas. We will use the trigonometric formula to reduce the equation in terms of known values.

Complete step by step answer:
We have to solve the equation $\dfrac{{\tan {{85}^ \circ } - \tan {{25}^ \circ }}}{{1 + \tan {{85}^ \circ }\tan {{25}^ \circ }}}$
$= \dfrac{{\tan {{85}^ \circ } - \tan {{25}^ \circ }}}{{1 + \tan {{85}^ \circ }\tan {{25}^ \circ }}}$
We will use the formula $\tan \left( {a - b} \right) = \dfrac{{\tan a - \tan b}}{{1 + \tan a\tan b}}$ to reduce the equation
$\Rightarrow \dfrac{{\tan a - \tan b}}{{1 + \tan a\tan b}} = \tan \left( {a - b} \right)$
We will take a as 85 and b equals to 25.
$\Rightarrow = \dfrac{{\tan {{85}^ \circ } - \tan {{25}^ \circ }}}{{1 + \tan {{85}^ \circ }\tan {{25}^ \circ }}} = \tan \left( {{{85}^ \circ } - {{25}^ \circ }} \right)$
$\Rightarrow = \dfrac{{\tan {{85}^ \circ } - \tan {{25}^ \circ }}}{{1 + \tan {{85}^ \circ }\tan {{25}^ \circ }}} = \tan \left( {{{60}^ \circ }} \right)$
We know that the value of tan60 is root 3
$\Rightarrow = \dfrac{{\tan {{85}^ \circ } - \tan {{25}^ \circ }}}{{1 + \tan {{85}^ \circ }\tan {{25}^ \circ }}} = \sqrt 3$
Hence, the value of equation $\dfrac{{\tan {{85}^ \circ } - \tan {{25}^ \circ }}}{{1 + \tan {{85}^ \circ }\tan {{25}^ \circ }}}$ is $\sqrt 3$.

Note: We should never directly put the values in the equation. We always try to reduce the equation using trigonometric formula like $\tan \left( {a - b} \right) = \dfrac{{\tan a - \tan b}}{{1 + \tan a\tan b}}$or some others. But in some easy level questions we can also find the solution just by putting the values.