Product Trigonometrical identity of Tangent and Cotangent

Trigonometric ratios tangent and cotangent are involved in a product relation by the reciprocal relation. The product relation between tangent and cotangent is true for every angle. Hence, the product relation between tangent and cotangent is called as a trigonometrical identity in product form or product trigonometric identity.

The trigonometric ratios tangent and cotangent are defined by the ratio of lengths of opposite side and adjacent side but their definitions are in reciprocal. The same reciprocal relation makes the product of tangent and cotangent at an angle is equal to one trigonometrically.

Proof

Triangle $CAB$ is a right angled triangle and it is having an angle theta ($\theta$). Mathematically, tangent and cotangent can be expressed by considering the triangle $CAB$.

$tan\theta$ $=$ $\frac{Length of opposite side}{Length of adjacent side}$ $=$ $\frac{BC}{AB}$

$cot\theta$ $=$ $\frac{Length of adjacent side}{Length of opposite side}$ $=$ $\frac{AB}{BC}$

Multiply the trigonometric ratios tangent and cotangent to get the product of them.

$tan\theta ×cot\theta =\frac{BC}{AB}×\frac{AB}{BC}$

The product of tangent and cotangent at an angle is equal to one.

Similarly,

$cot\theta ×tan\theta =1$

The product of cotangent and tangent at an angle is also equal to one.

The product of tangent and cotangent or product of cotangent and tangent is known as a trigonometric identity in product form.