Question

Prove the following identity :​

Answer 1

Answer:

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Answer 2

$\large\underline\textbf{Question :}$

$\text{Prove the following identity }$

$\large\mathtt{\frac{\cos\theta}{1-\sin\theta}+\frac{1-\sin\theta}{\cos\theta}=2\sec\theta}$

$\large\underline\textbf{Solution :}$

$\large\mathsf{\implies \frac{\cos\theta}{1-\sin\theta}+\frac{1-\sin\theta}{\cos\theta}=2\sec\theta}$

$\large\mathsf{\implies \frac{\cos^2\theta+{(1-\sin\theta)}^{2}}{2cos\theta(1-\sin\theta)}}$

$\large\mathsf{\implies \frac{{cos}^{2}\theta+1-2\sin\theta+{\sin}^{2}\theta}{\cos\theta(1-\sin\theta)}}$

$\color{Deeppink}{(\because {a-b}^{2}=a^2-2ab+b^2)}$

$\large\mathsf{\implies \frac{1+1-2\sin\theta}{\cos\theta(1-\sin\theta)}}$

$\color{deeppink}{(\because \sin^2\theta+\cos^2\theta=1)}$

$\large\mathsf{\implies \frac{2(1-\sin\theta)}{\cos\theta(1-\sin\theta)}}$

$\large\mathsf{\implies \frac{2}{\cos\theta}\implies2\times \frac{1}{\cos\theta} }$

$\large\mathsf{\implies 2\sec\theta }$

$\textbf{hence proved }$