Question

The value of $\tan \theta .\tan (90 - \theta )$is equals to:-
A. ${\sin ^2}\theta $
B. 1
C. ${\cos ^2}\theta $
D. 0

Answer 1

Hint: In order to solve this problem you need to use the basic trigonometry formulas like $\tan (90 - \theta ) = \cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }} = \dfrac{1}{{\tan \theta }}$. Using this will give you the right answer.

Complete step-by-step answer:

The given equation is $\tan \theta .\tan (90 - \theta )$…………………(1)
As we know that,$\tan (90 - \theta ) = \cot \theta $ …………………..(2)
And $\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}$ , $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$………………………..(3)
On putting the values in equation (1) which is obtained in (2) and (3) the new equation can be formulated as:
$\dfrac{{\sin \theta }}{{\cos \theta }} \times \dfrac{{\cos \theta }}{{\sin \theta }} = 1$
The terms will cancel out and the value will be 1.
Hence, the correct option is B.

Note: Whenever you face such types of problems you just need to use the basic identities of trigonometry and then solve the equation to get the value of the asked term. Here we have used the formulas $\tan (90 - \theta ) = \cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }} = \dfrac{1}{{\tan \theta }}$. Proceeding like this will provide you the correct solution.