Question

The expression \[\tan \left( 90-\theta \right)\] is equal to
(A) \[\cos \theta \]
(B) \[\sin \theta \]
(C) \[\left( \sin \theta +\cos \theta \right)\]
(D) \[\cot \theta \]

Answer 1

Hint: The given expression is \[\tan \left( 90-\theta \right)\] . We know that \[\tan \theta \] is the ratio of \[\sin \theta \] and \[\cos \theta \] , \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\] . Use this and modify the given expression as the ratio of sine function and cosine function. Now, we know the identity \[\sin \left( 90-\theta \right)=\cos \theta \] and \[\cos \left( 90-\theta \right)=\sin \theta \] . We also know that \[\cot \theta =\dfrac{\cos \theta }{\sin \theta }\] . Now, solve it further and get the value of the given expression.

Complete step by step answer:
According to the question, we are given a trigonometric expression and we have to solve it. Also, we are given four options and we have to pick the correct option after solving it.
The given expression = \[\tan \left( 90-\theta \right)\] ………………………………..(1)
We can observe that all the given four options in terms of \[\theta \] while the above expression is in terms of \[\left( 90-\theta \right)\] . So, the above expression needs to be simplified more.
We know that \[\tan \theta \] is the ratio of \[\sin \theta \] and \[\cos \theta \] , \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\] …………………………………(2)
 Now, on replacing \[\theta \] by \[\left( 90-\theta \right)\] in equation (2), we get
\[\tan \left( 90-\theta \right)=\dfrac{\sin \left( 90-\theta \right)}{\cos \left( 90-\theta \right)}\] ……………………………………(3)
Using equation (3) and on modifying equation (1), we get
The given expression = \[\tan \left( 90-\theta \right)=\dfrac{\sin \left( 90-\theta \right)}{\cos \left( 90-\theta \right)}\] …………………………………..(4)
We know the identity that \[\sin \left( 90-\theta \right)=\cos \theta \] …………………………………(5)
We also know the identity that \[\cos \left( 90-\theta \right)=\sin \theta \] …………………………………(6)
Now, from equation (4), equation (5), and equation (6), we get
\[\tan \left( 90-\theta \right)=\dfrac{\sin \left( 90-\theta \right)}{\cos \left( 90-\theta \right)}=\dfrac{\cos \theta }{\sin \theta }\] …………………………………………..(7)
 We know that \[\cot \theta \] is the ratio of \[\cos \theta \] and \[\sin \theta \] , \[\cot \theta =\dfrac{\cos \theta }{\sin \theta }\] …………………………………(8)
Now, from equation (7) and equation (8), we get
 \[\tan \left( 90-\theta \right)=\dfrac{\sin \left( 90-\theta \right)}{\cos \left( 90-\theta \right)}=\dfrac{\cos \theta }{\sin \theta }=\cot \theta \] …………………………………………..(9)
Therefore, the value of the given expression \[\tan \left( 90-\theta \right)\] is equal to \[\cot \theta \] .
Hence, the correct option is (D).

Note:
The best way for solving this type of question is to directly use the identity formula that \[\tan \left( 90-\theta \right)=\cot \theta \] . Using this formula saves time and reduces the complexity which might lead to calculation mistakes.