Question

If $(1 + {\tan ^2}\theta ).{\sin ^2}\theta =$
(a) ${\sin ^2}\theta$
(b) ${\cos ^2}\theta$
(c) ${\tan ^2}\theta$
(d) ${\cot ^2}\theta$

Answer 1

Hint : As we know that the above question is related to the trigonometry. $\sin ,\cos ,\tan ,\sec ,\cot$ and $\cos ec$ these are all trigonometric ratios. We will solve the above given question with the help of basic trigonometric formulas. We know that $\tan \theta$ can also be written in the form of $\dfrac{{\sin \theta }}{{\cos \theta }}$ . It is the reciprocal formula of trigonometric ratios.

Complete step-by-step answer :
As per the given question we have: $(1 + {\tan ^2}\theta ).{\sin ^2}\theta$ .
We have to find the value of the given expression by using the trigonometric formulas.
We will first take the left hand side of the equation and solve it by simplifying it in simple terms i.e.
$(1 + {\tan ^2}\theta ).{\sin ^2}\theta = \left( {1 + \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }}} \right) \times {\sin ^2}\theta$ .
By taking the L.C.M we can write it as
$(\dfrac{{{{\cos }^2}\theta + {{\sin }^2}\theta }}{{{{\cos }^2}\theta }}) \times {\sin ^2}\theta$ .
On further solving $\left( {\dfrac{1}{{{{\cos }^2}\theta }}} \right) \times {\sin ^2}\theta$ .
It gives us the value of ${\tan ^2}\theta$ which means $\dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }} = {\tan ^2}\theta$
Hence the required value is (c) ${\tan ^2}\theta$ .
So, the correct answer is “Option C”.

Note : We should note that the basic trigonometric formula is used in the above i.e. ${\sin ^2}\theta + {\cos ^2}\theta = 1$ . Trigonometric functions are also called circular functions and these basic functions are also known as trigonometric ratios. There are multiple trigonometric formulas and identities which represent the relation between the functions and enable to find the value of unknown angles.