Question

cosecO(secO - 1) - cotO(1 - cosO) =tanO - sinO​

Answer 1

Answer:

$$\begin{lgathered}LHS = sin\theta(1+ tan \theta) + cos\theta(1+cot\theta) \\= sin\theta\Big(1+\frac{ sin \theta}{cos \theta}\Big) + cos\theta\Big(1+\frac{cos\theta}{sin\theta}\Big)\\= sin\theta\Big(\frac{cos \theta+ sin \theta}{cos \theta}\Big) + cos\theta\Big(\frac{sin \theta+ cos\theta}{sin\theta}\Big)\\= (sin \theta + cos \theta) \Big (\frac{sin \theta}{cos\theta} + \frac{cos \theta}{sin \theta}\Big) \\= (sin \theta + cos \theta) \Big (\frac{sin^{2}\theta+ cos^{2} \theta}{cos \theta sin \theta}\Big) \\= (sin \theta + cos \theta) \Big (\frac{1}}{cos \theta sin \theta}\Big) \\= \frac{sin \theta + cos \theta }{ cos \theta sin \theta } \\= \frac{ sin \theta }{cos \theta sin \theta } + \frac{cos \theta }{cos \theta sin \theta } \\= \frac{1}{cos\theta} + \frac{1}{sin \theta} \\= sec \theta + Cosec \theta \\= RHS\end{lgathered}$$

$$Hence\: proved$$

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