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Implicit Differentiation

If tan theta ...

Question

If tan theta =a/b prove that,
a sin theta - b cos theta/ a sin theta +b cos theta = a²- b²/ a²+ b²

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Solution

$\tan θ = \frac{a}{b} To Prove : - \frac{a \sin θ - b \cos θ}{a \sin θ + b \cos θ} = \frac{a^{2} - b^{2}}{a^{2} + b^{2}} LHS \frac{a \sin θ - b \cos θ}{a \sin θ + b \cos θ} = \frac{\cos θ (\frac{a \sin θ}{\cos θ} - b)}{\cos θ (\frac{a \sin θ}{\cos θ} + b)} = \frac{a \tan θ - b}{a \tan θ + b} = \frac{a (\frac{a}{b}) - b}{a (\frac{a}{b}) + b} = \frac{\frac{a^{2} - b^{2}}{b}}{\frac{a^{2} + b^{2}}{b}} = \frac{a^{2} - b^{2}}{a^{2} + b^{2}} = RHS Hence Proved .$

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