Question

Find the value of the following without using tables:
$\tan {30^\circ } \times \tan {60^\circ } + \cos {0^\circ } + \sec {60^\circ }$

Answer 1

Hint: First convert $\tan {60^\circ }$ in terms of $\cot$ and then write $\cot {60^\circ }$ in terms of $\tan {30^\circ }$ to easily evaluate the first terms. Now write the value of $\cos {0^\circ }$ as it is and then convert $\sec$ into $\cos$ and then find the value of it. Here we must use tables for the last two terms because without which we cannot find the value of the expression.

Complete step by step solution:
The given expression is,
$\Rightarrow \tan {30^\circ } \times \tan {60^\circ } + \cos {0^\circ } + \sec {60^\circ }$
Firstly, convert $\tan {60^\circ }$ in terms of $\cot$.
We can do this by using the conversion, $\tan \theta = \dfrac{1}{{\cot \theta }}$
$\Rightarrow \tan {30^\circ } \times \dfrac{1}{{\cot {{60}^\circ }}} + \cos {0^\circ } + \sec {60^\circ }$
We shall now convert $\cot {60^\circ }$ to $\tan {30^\circ }$ to easily evaluate.
We can do this by using, $\cot \theta = \tan (90 - \theta )$
$\Rightarrow \tan {30^\circ } \times \dfrac{1}{{\tan ({{90}^\circ } - {{60}^\circ })}} + \cos {0^\circ } + \sec {60^\circ }$
On further evaluation,
$\Rightarrow \tan {30^\circ } \times \dfrac{1}{{\tan ({{30}^\circ })}} + \cos {0^\circ } + \sec {60^\circ }$
Simplify the first term to get,
$\Rightarrow 1 + \cos {0^\circ } + \sec {60^\circ }$
Write the value of $\cos {0^\circ }$ as it is since there is no further simplification in that term.
$\Rightarrow 1 + 1 + \sec {60^\circ }$
Now convert $\sec$ into $\cos$
$\Rightarrow 1 + 1 + \dfrac{1}{{\cos {{60}^\circ }}}$
Write the value of $\cos {60^\circ }$ in the expression and evaluate.
$\Rightarrow 1 + 1 + \dfrac{1}{{\dfrac{1}{2}}}$
$\Rightarrow 1 + 1 + 2$
On further evaluation we get,
$\Rightarrow 4$

$\therefore$ $\tan {30^\circ } \times \tan {60^\circ } + \cos {0^\circ } + \sec {60^\circ }$ is equal to the value $4$.

Additional Information: Whenever complex equations are given to solve one must always Firstly start from the complex side and then convert all the terms into $\cos \theta$ or $\sin \theta$ . Then combine them into single fractions. Now it’s most likely to use Trigonometric identities for the transformations if there are any. Know when and where to apply the Subtraction-Addition formula.

Note: Always check when the trigonometric functions are given in degrees or radians. There’s a lot of difference between both ${1^\circ } \times \dfrac{\pi }{{180}} = 0.017Rad$ . Express everything in $\sin \theta$ or $\sin \theta$ to easily evaluate. It is a must to memorize the values of basic trigonometric functions since all the functions can be written in terms of those basic trigonometric functions and can be easily evaluated.