Product Trigonometrical identity of Cosine and Secant

product relation of cosine and secant

Cosine is a trigonometrical ratio and has a product relation with another trigonometric ratio secant. Similarly, secant also has product relation with cosine. The product relation between cosine and secant is true for every angle. So, the product relation of cosine and secant is called as a product trigonometrical identity.

Cosine and secant are two trigonometrical ratios and they both are defined by the sides; hypotenuse and adjacent side but their definitions are in reciprocal. The reciprocal relation between cosine and secant makes their product is equal to one.

Proof:

ΔBAC is a right angled triangle and having an angle theta (θ). Trigonometrical ratios cosine and secant are expressed in mathematical form by considering sides and angle of the triangle.

right angled triangle

cos θ = Length of adjacent sideLength of hypotenuse = ABAC

sec θ = Length of hypotenuseLength of adjacent side = ACAB

Multiply trigonometrical ratios cosine and secant to get the product of them.

cos θ × sec θ = ABAC × ACAB

⇒   cos θ × sec θ = ABAB × ACAC

⇒   cos θ × sec θ = 1×1

∴   cos θ × sec θ = 1

The product of cosine and secant at an angle is equal to one.

Similarly,

sec θ × cos θ = 1

The product of secant and cosine at an angle is also equal to one.

The product relation between cosine and secant (or) secant and cosine is known as a trigonometrical identity in product form.

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