Prove that x 4 -y 4 =b 2 -a 2 ,if a cot teta+bcosec teta=x 2 and bcot teta+ acosec teta=y 2
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acotθ+bcosecθ=x²
bcotθ+acosecθ=y²
∴, x⁴-y⁴
=(x²)²-(y²)²
=a²cot²θ+2abcotθcosecθ+b²cosec²θ-b²cot²θ-2abcotθcosecθ-a²cosec²θ
=b²(cosec²θ-cot²θ)-a²(cosec²θ-cot²θ)
=b²-a² (Proved) [∵, cosec²θ-cot²θ=1]
bcotθ+acosecθ=y²
∴, x⁴-y⁴
=(x²)²-(y²)²
=a²cot²θ+2abcotθcosecθ+b²cosec²θ-b²cot²θ-2abcotθcosecθ-a²cosec²θ
=b²(cosec²θ-cot²θ)-a²(cosec²θ-cot²θ)
=b²-a² (Proved) [∵, cosec²θ-cot²θ=1]
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