A mutual product relation between two reciprocal trigonometrical ratios at an angle is called a product trigonometrical identity or trigonometric identity in product form.

In Trigonometry, a pair of trigonometrical ratios has mutual reciprocal relation and this reciprocal relation between them makes the product of them is equal to one. This product relation between trigonometrical ratios is used as formulas in problem solving and deriving new fundamentals in trigonometry.

The 6 trigonometrical ratios form three trigonometrical identities in product form by the reciprocal relation. Here, theta ($\theta $) is considered as an angle of a right angled triangle to express the product relations between two trigonometrical ratios in mathematical form.

1.

The trigonometrical ratios sine and cosecant both have reciprocal relation mutually because they are defined by the lengths of opposite side and hypotenuse of the right angled triangle but their definitions are in reciprocal.

Therefore, the product of sine and cosine is equal to one. Similarly, the product of cosine and sine is also equal to one.

$sin\theta \times csc\theta =1$ and $csc\theta \times sin\theta =1$

The product relation between sine and cosine is called as a trigonometrical identity in product form.

2.

The two trigonometric ratios Cosine and secant also have mutual reciprocal relation due to their fundamental definitions from the lengths of adjacent side and hypotenuse of the right angled triangle. However, their definitions are in reciprocal.

Hence, the product of cosine and secant equals to one and also the product of secant and cosine equals to one.

$cos\theta \times sec\theta =1$ and $sec\theta \times cos\theta =1$

The product relation between cosine and secant is called as a trigonometric identity in product form.

3.

Tangent and cotangent are two trigonometrical ratios and have reciprocal relation mutually because they both are defined in trigonometry by the lengths of opposite side and adjacent side but their definitions are in reciprocal.

The mutual reciprocal relation makes the product of tangent and cotangent is equals to one. Similarly, the product of cotangent and tangent is also equals to one by the same relation.

$tan\theta \times cot\theta =1$ and $cot\theta \times tan\theta =1$

The product relation between tangent and cotangent is another example for product form trigonometrical identity.

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