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If sinθ+cscθ=2\sin\theta + \csc\theta = 2, then what is the value of sin2θ+csc2θ\sin^2\theta + \csc^2\theta?

Solution

Correct Option: 3

Squaring both sides: (sinθ+cscθ)2=4(\sin\theta + \csc\theta)^2 = 4, so sin2θ+csc2θ+2sinθcscθ=4\sin^2\theta + \csc^2\theta + 2\sin\theta \cdot \csc\theta = 4.

Since sinθcscθ=1\sin\theta \cdot \csc\theta = 1: sin2θ+csc2θ=42=2\sin^2\theta + \csc^2\theta = 4 - 2 = 2.

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