Question

If ${\tan ^2}\theta = {\sin ^2}\theta + {\cos ^2}\theta $, then find the value of $\theta $ .
A. ${30^ \circ }$
B. ${45^ \circ }$
C. ${60^ \circ }$
D. ${90^ \circ }$

Answer 1

Hint: First we will use the identity we know that is ${\sin ^2}\theta + {\cos ^2}\theta = 1$. Then put this value in the given expression and after that take the square root of both the sides and choose the option among given ones.

Complete step-by-step answer:
Let us first see the identity, we are going to use to solve this question.
We know that: ${\sin ^2}\theta + {\cos ^2}\theta = 1$ for all $\theta \in \mathbb{R}$.
Putting this value in ${\tan ^2}\theta = {\sin ^2}\theta + {\cos ^2}\theta $.
We will now have the result as follows:-
$ \Rightarrow {\tan ^2}\theta = 1$ for all $\theta \in \mathbb{R}$.
Taking square root on both the sides, we will now have with us the following result:-
$ \Rightarrow \tan \theta = \sqrt 1 $
We know that ${(1)^2} = 1$ and ${( - 1)^2} = 1$.
Therefore, we have:-
$ \Rightarrow \tan \theta = \pm 1$
We know that tangent is positive in the first and third quadrant and negative in the second and fourth quadrant.
But if we look at the option, we have only options with first quadrant angles.
Therefore, tangents must take only positive values.
Therefore, the only possible value is:-
$ \Rightarrow \tan \theta = 1$
This is only possible when $\theta = {45^ \circ }$

So, the correct answer is “Option B”.

Note: The student might make the mistake of considering only the positive value while taking the square root whereas the options might have second or fourth quadrant values as well. So, you must check the options as well and consider every value until you get a solid reason to reject those values.
The students must also keep in mind that this might be a multiple correct question as well and you may have given the values of angles as per the value -1 as well. So, then you will need to choose every possible option.
The students must try to prove the identity used in this solution using the trigonometric values by drawing a right angled triangle and filling in the values, it will become extremely easy for you to remember it.