Question

What is the exact value of ${\tan ^2}45 + {\sec ^2}30$?

Answer 1

Here in this question, we have to find the exact value of a given trigonometric function. For this first, we have to know the table of standard degree values of trigonometric ratios and by this table we can take the value of ${\tan ^2}45$ and ${\sec ^2}30$ then by its square value and further simplify by using a basic arithmetic operations we get the required solution.

Complete step by step solution:
A function of an angle expressed as the ratio of two of the sides of a right triangle that contains that angle; the sine, cosine, tangent, cotangent, secant, or cosecant known as trigonometric function Also called circular function.
Consider the given question:
${\tan ^2}45 + {\sec ^2}30$ ------ (2)
As we know, from the standard angles table of trigonometric ratios the value of $\tan {45^ \circ } = 1$ and $\sec {30^ \circ } = \dfrac{2}{{\sqrt 3 }}$, then
Equation (1) becomes
$\Rightarrow \,\,\,{\left( 1 \right)^2} + {\left( {\dfrac{2}{{\sqrt 3 }}} \right)^2}$
By the quotient rule of exponent ${\left( {\dfrac{a}{b}} \right)^2} = \dfrac{{{a^2}}}{{{b^2}}}$, then we have
$\Rightarrow \,\,\,{\left( 1 \right)^2} + \dfrac{{{2^2}}}{{{{\left( {\sqrt 3 } \right)}^2}}}$
$\Rightarrow \,\,\,1 + \dfrac{4}{3}$
Take 3 as LCM
$\Rightarrow \,\,\,\dfrac{{3 + 4}}{3}$
$\Rightarrow \,\,\,\dfrac{7}{3}$
On simplification, we get
$\therefore \,\,\,\,2.333333$
Hence, the exact value of ${\tan ^2}45 + {\sec ^2}30$ is $2.333333$.

Note:
When solving the trigonometry-based questions, we have to know the definitions and table of standard angles of all six trigonometric ratios. Remember, the table of value of standard angles ${0^ \circ }$, ${30^ \circ }$, ${45^ \circ }$, ${60^ \circ }$ and ${90^ \circ }$ of all six trigonometric ratios and should know the identities, double angle, half angle, sum identity and difference identity of trigonometric ratios it makes the solution more easier.