# Proof for Cosine by Sine Quotient identity

According to the quotient of cosine by sine identity, the quotient of cosine by sine is equal to the cotangent.

$\dfrac{\cos{\theta}}{\sin{\theta}} \,=\, \cot{\theta}$

The trigonometric quotient identity is derived geometrically in mathematical form from a right triangle. Now, it’s your time to learn how to prove the quotient of cosine by sine identity from a geometric shape right triangle.

$\Delta BAC$ is a right angled triangle. In this triangle, $\overline{BC}$, $\overline{AB}$ and $\overline{AC}$ are opposite side, adjacent side and hypotenuse respectively, and the angle of this triangle is taken as theta.

### Define Cosine and Sine functions

According to the definitions of the trigonometric functions, express the sine and cosine functions in ratio form at an angle theta.

$\sin{\theta} \,=\, \dfrac{BC}{AC}$

$\cos{\theta} \,=\, \dfrac{AB}{AC}$

### Divide the Cosine by Sine function

Now, divide the cosine function by the sine function for evaluating the quotient of them.

$\dfrac{\cos{\theta}}{\sin{\theta}} \,=\, \dfrac{\dfrac{AB}{AC}}{\dfrac{BC}{AC}}$

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{AC} \times \dfrac{AC}{BC}$

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{BC} \times \dfrac{AC}{AC}$

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{BC} \times \require{cancel} \dfrac{\cancel{AC}}{\cancel{AC}}$

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{BC} \times 1$

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{BC}$

### Express the Ratio in Trigonometric function

We have derived in the above step that the ratio of cosine by sine is equal to the quotient of lengths of adjacent side ($AB$) by opposite side ($BC$).

$\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{BC}$

According to the trigonometry, the quotient of $AB$ by $BC$ represents the cotangent. Here, the angle of right triangle is theta. Therefore, the cot function is written as $\cot{\theta}$.

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\cot{\theta}$

Therefore, it is proved that the quotient of cosine function by sine function is equal to the cotangent function mathematically.

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