Find the value of sin2 θ + cosec2 θ.

If sin θ + cosec θ = 3 1/3, find the value of sin² θ + cosec² θ.

Answer :

sin θ + cosec θ = 3 1/3 = 10/3

By squaring on both sides

(sin θ + cosec θ)² = (10/3)²

Expanding using formula (a + b)² = a² + b² + 2ab

sin² θ + cosec² θ + 2 sin θ cosec θ = 100/9

We know that sin θ = 1/cosec θ

sin² θ + cosec² θ + (2 sinθ× 1/sin θ) = 100/9

sin² θ + cosec² θ + 2 = 100/9

⇒ sin² θ + cosec² θ = 100/9 – 2

Taking LCM

sin² θ + cosec² θ = (100 – 18)/9 = 82/9

sin² θ + cosec² θ = 9 1/9

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