# Simplify the following algebraic term:

$

\left( {1 + {{\tan }^2}\theta } \right).{\sin ^2}\theta = \\

A.{\text{ }}{\sin ^2}\theta \\

B.{\text{ }}{\cos ^2}\theta \\

C.{\text{ }}{\tan ^2}\theta \\

D.{\text{ }}{\cot ^2}\theta \\

$

Last updated date: 26th Mar 2023

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Answer

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310.8k+ views

Hint- For solving, use simple trigonometric identities and formulas.

Since we know that ${\text{se}}{{\text{c}}^2}\theta - {\tan ^2}\theta = 1$

$ \Rightarrow {\text{se}}{{\text{c}}^2}\theta = 1 + {\tan ^2}\theta $

Substituting the above equation in the give question,

Now the question becomes$\left[ {{\text{se}}{{\text{c}}^2}\theta } \right].{\sin ^2}\theta $

Further simplifying the trigonometric terms

$

\Rightarrow {\left[ {\sec \theta .\sin \theta } \right]^2} \\

\Rightarrow {\left[ {\dfrac{1}{{\cos \theta }}.\sin \theta } \right]^2} \\

\Rightarrow {\left[ {\tan \theta } \right]^2}\because \dfrac{{\sin \theta }}{{\cos \theta }} = \tan \theta \\

\Rightarrow {\tan ^2}\theta \\

$

Hence, the correct option is $C$

Note- Use Simple trigonometric formulas which are mentioned above along with the solution. These formulas must be remembered. Always try to reduce the equation by the use of trigonometric identities which further reduces the equation.

Since we know that ${\text{se}}{{\text{c}}^2}\theta - {\tan ^2}\theta = 1$

$ \Rightarrow {\text{se}}{{\text{c}}^2}\theta = 1 + {\tan ^2}\theta $

Substituting the above equation in the give question,

Now the question becomes$\left[ {{\text{se}}{{\text{c}}^2}\theta } \right].{\sin ^2}\theta $

Further simplifying the trigonometric terms

$

\Rightarrow {\left[ {\sec \theta .\sin \theta } \right]^2} \\

\Rightarrow {\left[ {\dfrac{1}{{\cos \theta }}.\sin \theta } \right]^2} \\

\Rightarrow {\left[ {\tan \theta } \right]^2}\because \dfrac{{\sin \theta }}{{\cos \theta }} = \tan \theta \\

\Rightarrow {\tan ^2}\theta \\

$

Hence, the correct option is $C$

Note- Use Simple trigonometric formulas which are mentioned above along with the solution. These formulas must be remembered. Always try to reduce the equation by the use of trigonometric identities which further reduces the equation.

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