Question

What are the quotient identities for trigonometric functions?

Answer 1

Hint: Now the quotient identities are nothing but the trigonometric functions which are ratios of sin and cos. Quotient identities are nothing but the relation of tan and cot in terms of sin and cos. Hence we will use the definition to identify the relation between sin, tan and cos.

Complete step-by-step solution:
Now first let us understand the trigonometric function.
Trigonometric functions are nothing but the constant ratios we get in a right angle triangle. There are 6 trigonometric ratios namely sin, cos, tan, cot, sec and cosec.
Now sin is the ratio of the opposite side and hypotenuse in the right angle triange.
Similarly cos is the ratio of the adjacent side and hypotenuse in the right angle triangle.
Now tan is the ratio of the opposite side and adjacent side of the right angle triangle.
Now the functions cot, sec and cosec are nothing but reciprocals of the ratios tan, cos and sin respectively.
Now we know that $\text{sin}\theta \text{=}{\displaystyle \frac{\text{opposite side}}{\text{hypotenuse}}}$ ,$\text{cos}\theta \text{=}{\displaystyle \frac{\text{adjecent side}}{\text{hypotenuse}}}$ and $\text{tan}\theta \text{=}{\displaystyle \frac{\text{opposite side}}{\text{adjecent side}}}$ .
Hence we can clearly see that $\tan \theta ={\displaystyle \frac{\sin \theta }{\cos \theta }}$ .
Now since cot is nothing but reciprocal of tan we have $\cot \theta ={\displaystyle \frac{\text{cos}\theta }{\text{sin}\theta }}$
Now these two ratios tan and cot are known as quotient identities.

Note: Now in trigonometric we also have three major identities and Pythagoras identities. The identities states ${\sin }^2\theta +{\cos }^2\theta =1$, $1+{\cot }^2\theta ={\csc }^2\theta$ and $1+{\tan }^2\theta ={\sec }^2\theta$ . These identities are obtained by Pythagoras theorem of right angles triangle which states $a^2+b^2=c^2$ where a and b are perpendicular sides and c is the hypotenuse of triangle.