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Question
Prove that `sin^2 θ/ cos^2 θ + cos^2 θ/sin^2 θ = 1/(sin^2 θ. cos^2 θ) - 2`.
Solution
LHS = `sin^2 θ/ cos^2 θ + cos^2 θ/sin^2 θ`
= `(sin^4θ + cos^4θ)/(sin^2θ.cos^2θ)`
= `((sin^2 θ + cos^2 θ)^2 - 2(sin^2 θ. cos^2 θ))/(sin^2 θ.cos^2 θ)`
= `((1)^2 - 2sin^2θ. cos^2 θ)/(sin^2 θ.cos^2 θ)`
= `1/(sin^2 θ.cos^2 θ) - (2sin^2θ. cos^2 θ)/(sin^2 θ.cos^2 θ)`
= `1/(sin^2 θ.cos^2 θ) - 2`
= RHS
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