### Theory:

Based on the three basic trigonometric ratios $$\sin$$, $$\cos$$ and $$\tan$$ we will define its reciprocals.
Reciprocal Ratios:
Consider a right-angled triangle with a corresponding angle $$\theta$$.

The three basic trigonometric ratios are:

• Sine
• Cosine
• Tangent

The table below depicts the relation of reciprocal ratios with the right-angled triangle.

 Name of the angle Sine Cosine Tangent Short form of the angle $$\sin$$ $$\cos$$ $$\tan$$ Relationship $$\sin \theta$$ $$=$$ $$\frac{\text{Opposite side}}{\text{Hypotenuse}}$$ $$\cos \theta$$ $$=$$ $$\frac{\text{Adjacent side}}{\text{Hypotenuse}}$$ $$\tan \theta$$ $$=$$ $$\frac{\text{Opposite side}}{\text{Adjacent side}}$$ Name of the reciprocal angle Cosecant Secant Cotangent Short form of the angle $$\text{cosec}$$ $$\sec$$ $$\cot$$ Measurements related to the right-angled triangle Relationship $$\text{cosec}\,\theta$$ $$=$$ $$\frac{\text{Hypotenuse}}{\text{Opposite side}}$$ $$\sec \theta$$ $$=$$ $$\frac{\text{Hypotenuse}}{\text{Adjacent side}}$$ $$\cot \theta$$ $$=$$ $$\frac{\text{Adjacent side}}{\text{Opposite side}}$$ Relation with the basic ratio $$\text{cosec}\,\theta$$ $$=$$ $$\frac{1}{\sin \theta}$$ or $$\sin \theta$$ $$=$$ $$\frac{1}{\text{cosec}\,\theta}$$ $$\sec \theta$$ $$=$$ $$\frac{1}{\cos \theta}$$ or $$\cos \theta$$ $$=$$ $$\frac{1}{\sec \theta}$$ $$\cot \theta$$ $$=$$ $$\frac{1}{\tan \theta}$$ or $$\tan \theta$$ $$=$$ $$\frac{1}{\cot \theta}$$

We can write certain identities based on these relationships.

• Identity $$1$$:
$$\text{cosec}\,\theta \times \sin \theta$$ $$=$$ $$\text{cosec}\,\theta \times \frac{1}{\text{cosec}\,\theta}$$

$$=$$ $$1$$

Therefore, $$\text{cosec}\,\theta \cdot \sin \theta= 1$$.
• Identity $$2$$:
$$\sec \theta \times \cos \theta$$ $$=$$ $$\sec \theta \times \frac{1}{\sec \theta}$$

$$=$$ $$1$$

Therefore, $$\sec \theta \cdot \cos \theta = 1$$.
• Identity $$3$$:
$$\cot \theta \times \tan \theta$$ $$=$$ $$\cot \theta \times \frac{1}{\cot \theta}$$

$$=$$ $$1$$

Therefore, $$\cot \theta \cdot \tan \theta = 1$$.