Chapter 5, Problem 137RE

To determine

Whether any trigonometric equation which has an infinite number of solutions is an identity or not.

Answer to Problem 137RE

Any trigonometric equation which has an infinite number of solutions is not an identity.

Explanation of Solution

Formula used:

The Pythagorean identity:

  ${\sin }^2\theta +{\cos }^2\theta =1$

The general solution of the trigonometric equation $\tan x=y$ is,

  $x=n\pi +{\tan }^{-1}\left(y\right)$

Know that if a trigonometric equation has an infinite number of solutions then it is not necessary that the equation is an identity.

Consider a trigonometric equation

  $\sin x-\cos x=0$

Write the above equation as:

  $\begin{array}{c}\sin x-\cos x=0\\ \sin x=\cos x\\ \frac{\sin x}{\cos x}=\frac{\cos x}{\cos x}\\ \tan x=1\end{array}$

It is known that the general solution of the trigonometric equation $\tan x=y$ is,

  $x=n\pi +{\tan }^{-1}\left(y\right)$

So, for the equation $\tan x=1$ , the general solution will be $x=n\pi +{\tan }^{-1}\left(1\right)$ .

Here, ${\tan }^{-1}\left(1\right)=\frac{\pi }{4}$

Substitute $\frac{\pi }{4}$ for ${\tan }^{-1}\left(1\right)$ in the solution $x=n\pi +{\tan }^{-1}\left(1\right)$ .

  $x=n\pi +\frac{\pi }{4}$

Thus, the general solution of the trigonometric equation $\sin x-\cos x=0$ has an infinite number of solutions; but the equation is not an identity.

Hence, any trigonometric equation which has an infinite number of solutions is not an identity.