Chapter 5, Problem 23RE

Solution

To find: The given equation $\sec x-\sin x\tan x=\cos x$ is likely to be an identity.

It is proved that the given equation $\sec x-\sin x\tan x=\cos x$ is likely to be an identity.

Given information:

The given equation $\sec x-\sin x\tan x=\cos x$ is likely to be an identity.

Formula used:

The formula of Pythagorean identities are $1+{\tan }^2\theta ={\sec }^2\theta$ and ${\cot }^2\theta +1={\csc }^2\theta$ .

Calculation:

Apply basic identities of trigonometric functions, $\sec \theta =\frac{1}{\cos \theta }$ and $\tan \theta =\frac{\sin \theta }{\cos \theta }$ into the left side of the given expression and simplify.

  $\begin{array}{c}\sec x-\sin x\tan x=\frac{1}{\cos x}-\sin x\left(\frac{\sin x}{\cos x}\right)\\ =\frac{1}{\cos x}-\frac{{\sin }^{2}x}{\cos x}\\ =\frac{1-{\sin }^{2}x}{\cos x}\end{array}$

Apply the formula of Pythagorean identities, $1-{\sin }^2\theta ={\cos }^2\theta$ into obtained expression and simplify.

  $\begin{array}{c}\frac{1-{\sin }^{2}x}{\cos x}=\frac{{\cos }^{2}x}{\cos x}\\ =\cos x\end{array}$

The obtained expression is same as the given right side expression. So it is proved that the given equation is likely to be an identity.

Draw the graph of the functions $y=\sec x-\sin x\tan x$ and $y=\cos x$ on the graphing calculator.

  

The graph of the function $y=\cos x$ is shown below.

  

Both the obtained graphs are similar and appear to be identical.

Thus, it can be said that the given equation is likely to be an identity from graphical support also.