Question

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

Answer 1

Hint: 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.