Chapter 5.1, Problem 78E

Solution

To find: Which of the given option is correct.

The correct option is (A).

Given information:

It is asked about the four of the six basic trigonometric functions.

Formula used:

The function is said to be even if it follows $f\left(-x\right)=f\left(x\right)$ and the function is said to be odd if it follows $f\left(-x\right)=-f\left(x\right)$ .

Calculation:

From the Even-odd Identity,

  $\begin{array}{c}\sin \left(-x\right)=-\sin x\\ \csc \left(-x\right)=-\csc x\\ \tan \left(-x\right)=-\tan x\\ \cot \left(-x\right)=-\cot x\end{array}$

So, these four identities are odd.

The other two identities are,

  $\begin{array}{c}\cos \left(-x\right)=\cos x\\ \sec \left(-x\right)=\sec x\end{array}$

So, these two identities are even.

Therefore, exactly four of the six basic trigonometric functions are “odd.”