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Question

If sin θ = 4553, find the value of cosec2 θ - cot2 θ

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Solution

Given:
sin θ = 4553
We know that sin θ = perpendicularhypotenuse= 4553
Let perpendicular = 45k and hypotenuse = 53k, where k is a positive constant
In the right triangle, we have:

Hypotenuse2 = perpendicular2+ base2 53k2 = 45k2+base22809k2 = 2025k2+ base2Base2 = 784 k2Base = ±28kBase = +28k ( Length cannot be negative)Now, we have:cosec θ = hypotenuseperpendicular= 5345cot θ= BasePerpendicular= 2845

Now, cosec2θ-cot2θ = 53452- 28452= 532-282452 = 20252025 = 1

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