Without using tables, evaluate:
$4\tan {60^0}\sec {30^0} + \dfrac{{\sin {{31}^0}\sec {{59}^0} + \cot {{59}^0}\cot {{31}^0}}}{{8{{\sin }^2}{{30}^0} - {{\tan }^2}{{45}^0}}}$

Hint: We have given a trigonometric expression. We have to find its value without using table value. So firstly we use some trigonometric identities to simplify the expression that trigonometric identities help us cancel the similar ferries in divide. After getting a simplified expression. We can put values of known angles and get the result.

Complete answer:
We have given a trigonometric expression
$\Rightarrow$$4\tan {60^0}\sec {30^0} + \dfrac{{\sin {{31}^0}\sec {{59}^0} + \cot {{59}^0}\cot {{31}^0}}}{{8{{\sin }^2}{{30}^0} - {{\tan }^2}{{45}^0}}}$
We have to solve this without using the table values.
Now we know that $\sin (90 - \theta )$ is equal to $\cos \theta $.
Also $\tan (90 - \theta )$ is equal to cot$\theta $.
and $sec(90 - \theta )$ is equal to cot$\theta $.
Therefore the express become by applying these formula
$\Rightarrow$$4\tan {60^0}\sec {30^0} + \dfrac{{\sin {{31}^0}{\text{ cosec3}}{{\text{1}}^0} + \cot {{59}^0}{\text{ tan5}}{{\text{9}}^0}}}{{8{{\sin }^2}{{30}^0} - {{\tan }^2}{{45}^0}}}$
Also we know that $\sin \theta $ is equal to $\dfrac{1}{{\cos ec \theta }}$
So the expression become
$\Rightarrow$$4\tan {60^0}\sec {30^0} + \dfrac{{\dfrac{1}{{\cos ec{{31}^0}}} \times \cos ec{{31}^0} + \dfrac{1}{{\tan {{59}^0}}} \times \tan {{59}^0}}}{{8{{\sin }^2}{{30}^0} - {{\tan }^2}{{45}^0}}}$
$ = 4\tan {60^0}\sec {30^0} + \dfrac{{1 + 1}}{{8{{\sin }^2}{{30}^0} - {{\tan }^2}{{45}^0}}}{\text{ }}................{\text{(i)}}$
Now value of $\tan {60^0}$ is equal to $\sqrt 3 $ value of $\sec {30^0}$ is equal to $\dfrac{2}{{\sqrt 3 }}$
Value of $\sec {30^0}$ is equal to $\dfrac{1}{2}$.
Value of $\tan {45^0}$ is equal to $1$ .

Putting all these values in the equation (i)

$\Rightarrow$\[4 \times \left( {\sqrt 3 } \right) \times \dfrac{2}{{\sqrt 3 }} + \dfrac{2}{{8 \times {{\left( {\dfrac{1}{2}} \right)}^2} - {{\left( 1 \right)}^2}}}\]
Simplifying the expression
$\Rightarrow$\[4 \times 2 + \dfrac{2}{{8 \times \dfrac{1}{4} - 1}}\]
$\Rightarrow$\[8 + \dfrac{2}{{2 - 1}}{\text{ }} \Rightarrow 8 + 2 = 10\]

So, the value of the expression is equal to $10$.

Note: Trigonometric is the branch of mathematics that studies the relationship between side lengths and angles of the triangle. Trigonometry has six trigonometric functions. Which are $\sin {\text{, cos, tan, cosec, sec and cot}}$. Trigonometric functions are the real functions which relate an angle of right angle triangles to the ratio of two sides of a triangle. Trigonometric functions are also called circular functions. With the help of these trigonometric functions we can drive lots of trigonometric formulas.