Question

If $\tan 45{}^\circ =\cot \theta $ , then value of $\theta $ in radian is
a)$\pi $
b)$\dfrac{\pi }{9}$
c)$\dfrac{\pi }{4}$
d)$\dfrac{\pi }{12}$

Answer 1

Hint: When an equation is given in terms of sine, cosine, tangent, we must use any of the trigonometric identities to make the inequation solvable. There are inter relations between sine, cosine, tan, secant these are inter relations are called as identities. Whenever you can see conditions such that \[\theta \in R\] , that means inequality is true for all angles. So, directly think of identity which will make your work easy.

\[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\] for all \[\theta \] , \[\tan \left( A+B \right)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}\]

Complete step-by-step answer:
An equality with sine, cosine or tangent in them is called a trigonometric equation. These are solved by some inter relations known beforehand.

All the inter relations which relate sine, cosine, tangent, secant, cotangent are called trigonometric identities. These trigonometric identities solve the equation and make them simpler to understand for a proof. These are the main and crucial steps to take us nearer to result.

Given equation in the question which we use here is:

$\tan 45{}^\circ =\cot \theta $

Let the value of $\tan 45{}^\circ $ is denoted by A, we get

$A=\tan 45{}^\circ \text{ }\Rightarrow \text{cot}\theta \text{=}A$

By basic knowledge of trigonometry, we can say the values as:

$\tan 45{}^\circ =1$

By above we can say the values of A to be as:

$A=1$

By this we say the value of cotangent to be as:

$\cot \theta =1$

By applying the ${{\cot }^{-1}}$ on both sides, we turn the equation to:

${{\cot }^{-1}}\left( \cot \theta \right)={{\cot }^{-1}}\left( 1 \right)$

$\theta =45{}^\circ $

By converting it into radians by multiplying with $\dfrac{\pi }{180}$ , we get:

$\theta =45\times \dfrac{\pi }{180}$

By simplifying the above, we get the value of angle as:

$\theta =\dfrac{\pi }{4}$

Option (c) is the correct answer.

Note: Be careful while using these inverse conditions, as when you apply inverse you get more than 1 solution. In that place you must check options for the right answer among them.