Question

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 \\
 $

Answer 1

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.