Proof of Reciprocal Identities

Now, that we know the reciprocal identities of trigonometry, let us now prove each one of them using the definition of the basic trigonometric functions. First, we will derive the reciprocal identity of the sine function. Consider a right-angled triangle ABC with a right angle at C.

We know that sin θ = Perpendicular/Hypotenuse = c/a and cosec θ = Hypotenuse/Perpendicular = a/c ⇒ sin θ is the reciprocal of cosec θ and cosec θ is the reciprocal of sin θ. Similarly, we will prove other reciprocal identities. cos θ = Base/Hypotenuse = b/a and cosec θ = Hypotenuse/Base = a/b ⇒ cos θ is the reciprocal of sec θ and sec θ is the reciprocal of cos θ. tan θ = sin θ/cos θ and cot θ = cos θ/sin θ ⇒ tan θ is the reciprocal of cot θ and cot θ is the reciprocal of tan θ. Hence, we have

• sin θ is the reciprocal of cosec θ
• cosec θ is the reciprocal of sin θ
• cos θ is the reciprocal of sec θ
• sec θ is the reciprocal of cos θ
• tan θ is the reciprocal of cot θ
• cot θ is the reciprocal of tan θ