Question

Solve the equation given below
\[\tan {10^0}\tan {15^0}\tan {75^0}\tan {80^0}\]

Answer 1

Hint-To solve, arrange the angles whose sum is equal to 90 and make use of the property of tangent.

$\tan ({90^0} - \theta ) = \cot \theta $
The given equation is
\[ = \tan {10^0}\tan {15^0}\tan {75^0}\tan {80^0}\]
Now arrange the terms whose sum of angle is equal to ${90^0}$
$ = (\tan {10^ 0 }\tan {80^0 })(\tan {75^0 }\tan {15^0 })$
As we know that $\left[ {\tan ({{90}^0 } - \theta ) = \cot \theta } \right]$
$
= (\tan ({90^0} - {80^0})\tan {80^0})(\tan ({90^0} - {15^0})\tan {15^0}) \\
= (\cot {80^0}\tan {80^0})(\cot {15^0}\tan {15^0}) \\
$
Again using the property of tangent
$\left[ {\tan \theta = \dfrac{1}{{\cot \theta }}} \right]$
$ = (\dfrac{1}{{\tan {{80}^0}}} \times \tan {80^0})(\dfrac{1}{{\tan {{15}^0}}} \times \tan {15^0})
\\
= 1 \\
$

Note- For solving these types of problems you must remember all the trigonometric function
expressions and their values. There are different approaches to solve these types of questions, one approach is to convert the given equation in a single variable and then solve. But in this question expression is already given in terms of tangent, so we arranged them in terms of angles whose sum is 90 and then applied the formula. Similarly in other questions we have to do these types of manipulations.