# Quotient Trigonometric identity of Cosine and Sine with Cotangent

## Definition

The relation of ratio of cosine to sine with cotangent at an angle is called quotient trigonometric identity of cosine and sine with cotangent.

Cosine, sine and cotangent are three trigonometric ratios in trigonometry and the three trigonometric ratios have quotient relation. The quotient of cosine and sine has a direct relation with cotangent. In other words, the ratio of cosine to sine is equal to cotangent.

The relation between three of them is used as a formula in mathematics to express the ratio of cosine to sine as cotangent and vice-versa.

### Proof

$\Delta BAC$ is a right angled triangle and the angle of right angled triangle is theta $(\theta)$.

$$\cos \theta = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse}$$

$$\sin \theta = \frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$$

Divide trigonometric function cosine by sine to obtain the quotient of them.

$$\frac{\cos \theta}{\sin \theta} = \frac{\frac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse}}{\frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}}$$

$$\Rightarrow \frac{\cos \theta}{\sin \theta} = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse} \times \frac{Length \, of \, Hypotenuse}{Length \, of \, Opposite \, side}$$

$$\Rightarrow \frac{\cos \theta}{\sin \theta} = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Opposite \, side} \times \frac{Length \, of \, Hypotenuse}{Length \, of \, Hypotenuse}$$

$$\Rightarrow \frac{\cos \theta}{\sin \theta} = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Opposite \, side}$$

As per $\Delta BAC$

$$\frac{Length \, of \, Adjacent \, side}{Length \, of \, Opposite \, side} = \cot \theta$$

Substitute this ratio in the above expression to obtain the relation of cosine and sine in terms of cotangent.

$$\frac{\cos \theta}{\sin \theta} = \cot \theta$$

It is proved that the value of ratio of cosine to sine is equal to cotangent.

### Verification

Take angle of right angled triangle $CAB$ is $45^°$.

Check Left hand side trigonometric expression by substituting $\theta$ by $45^°$.

$$\frac{\cos \theta}{\sin \theta} = \frac{\cos 45^°}{\sin 45^°}$$

$$\Rightarrow \frac{\cos 45^°}{\sin 45^°} = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}$$

$$\Rightarrow \frac{\cos 45^°}{\sin 45^°} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{1}$$

$$\Rightarrow \frac{\cos 45^°}{\sin 45^°} = 1$$

Check the right hand side trigonometric expression by substituting same angle.

$\tan \theta = \tan 45^°$

$\Rightarrow \tan 45^° = 1$

The quotient of ratio of cosine to sine at angle $45^°$ is $1$ and the value of cotangent at angle $45^°$is also equal to $1$. It is proved in the case of angle $45^°$ and it also proved for every angle. Hence, it is called as a trigonometric identity.

Due to the formation of this identity by the quotation of the ratio of trigonometric functions cosine to sine, the quotient relation of cosine to sine with cotangent is called as quotient trigonometric identity.

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