Find square root of -4 + root( 8 + 16cosec4 theta + sin4 theta)

We have,8+16cosec4θ+sin4θ=sin4θ+8+16cosec4θ=sin2θ2+2×sin2θ×4cosec2θ+4cosec2θ2=sin2θ+4cosec2θ28+16cosec4θ+sin4θ=sin2θ+4cosec2θ2=sin2θ+4cosec2θNow,-4+8+16cosec4θ+sin4θ=-4+sin2θ+4cosec2θ=sin2θ-4+4cosec2θ=sinθ2-2×sinθ×2cosecθ+2cosecθ2=sinθ-2cosecθ2=sinθ-2cosecθ

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