# Product Trigonometrical identity of Sine and Cosecant

Trigonometric ratio sine has product relation with another trigonometric ratio cosecant and cosecant also has product relation with same sine. The product relation between them is true for all the angles. Therefore, the product relation between sine and cosecant is called as a product trigonometrical identity.

Trigonometrically, the ratio of lengths of opposite side to hypotenuse is sine and the ratio of lengths of hypotenuse to opposite side is cosecant. The trigonometrical ratios are defined by the same sides but the definitions of sine and cosecant are in reciprocal. This reciprocal relation makes the product of sine and cosine equals to 1.

## Proof:

Triangle $BAC$ is a right angled triangle and its angle is theta ($\theta$). Sine and cosecant are expressed in mathematical from by considering this triangle.

$sin\theta$ $=$ $\frac{Length of opposite side}{Length of hypotenuse}$ $=$ $\frac{BC}{AC}$

$csc\theta$ $=$ $\frac{Length of hypotenuse}{Length of opposite side}$ $=$ $\frac{AC}{BC}$

Multiply sine and cosine to get their product

$sin\theta ×csc\theta =\frac{BC}{AC}×\frac{AC}{BC}$

The product of sine and cosecant is one.

Similarly,

$csc\theta ×sin\theta =1$

The product of cosecant and sine is also one.

Therefore, the product relation between sine and cosecant or cosecant and sine is known as a product trigonometric identity.

Save (or) Share