91Ó°ÊÓ

Here we have to prove the following quotient identities :-

(a) $\tan \mathrm{θ}=\frac{\sin \mathrm{θ}}{\cos \mathrm{θ}}$

(b) $cot\mathrm{θ}=\frac{\cos \mathrm{θ}}{\sin \mathrm{θ}}$

We will apply trigonometry ratios, to prove these quotient identities.

Consider the following right angled triangle.

For this right angled triangle, sine function is defined as perpendicular upon hypotenuse.

That is :-

$\sin \mathrm{θ}=\frac{AB}{AC}$$..........\left(1\right)$

Also cosine function is defined as base upon hypotenuse.

That is :-

localid="1647181073885" $\cos \mathrm{θ}=\frac{BC}{AC}$$.........\left(2\right)$

Now tangent function is defined as perpendicular upon base.

That is :-

$\tan \mathrm{θ}=\frac{AB}{BC}$$.........\left(3\right)$

Divide $\left(1\right)by\left(2\right)$, then we have :-

role="math" localid="1647181511822" $$\begin{gathered} \frac{\sin \mathrm{θ}}{\cos \mathrm{θ}}=\frac{{\displaystyle \frac{AB}{AC}}}{{\displaystyle \frac{BC}{AC}}} \\ ⇒\frac{\sin \mathrm{θ}}{\cos \mathrm{θ}}=\frac{AB}{AC}×\frac{AC}{BC} \\ ⇒\frac{\sin \mathrm{θ}}{\cos \mathrm{θ}}=\frac{AB}{BC}............\left(3\right) \end{gathered}$$

By comparing $\left(2\right)and\left(3\right)$, we have :-

The right hand sides of both equations are equal, so left hand sides are also equal. This gives us :-

$\tan \mathrm{θ}=\frac{\sin \mathrm{θ}}{\cos \mathrm{θ}}$

Hence proved.

Consider the following right angled triangle.

For this right angled triangle, cotangent function is defined as base upon perpendicular.

That is :-

$cot\mathrm{θ}=\frac{BC}{AB}$$.........\left(4\right)$

Divide $\left(2\right)by\left(1\right)$, then we have :-

localid="1647182332148" $$\begin{gathered} \frac{\cos \mathrm{θ}}{\sin \mathrm{θ}}=\frac{{\displaystyle \frac{BC}{AC}}}{{\displaystyle \frac{AB}{AC}}} \\ ⇒\frac{\cos \mathrm{θ}}{\sin \mathrm{θ}}=\frac{BC}{AC}×\frac{AC}{AB} \\ ⇒\frac{\cos \mathrm{θ}}{\sin \mathrm{θ}}=\frac{BC}{AB}...........\left(5\right) \end{gathered}$$

Now by comparing $\left(4\right)and\left(5\right)$, we have :-

The right hand sides of both equations are equal, so left hand sides are also equal. This gives us :-

$cot\mathrm{θ}=\frac{\cos \mathrm{θ}}{\sin \mathrm{θ}}$

Hence proved.