Question

$\frac{ \cos \theta}{1 + \sin \theta} + \frac{1 + \sin \theta}{cos \theta} = 2sec \theta$
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Answer 1

$\Huge question$

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

$\Huge solution$

First we simplify LHS

$\large LHS = \small \frac{ \cos \theta}{1 + \sin \theta} + \frac{1 + \sin \theta}{ \cos \theta} \\ \implies \ \frac{ { \cos}^{2} \theta + {(1 + \sin \theta)}^{2} }{(1 + \sin \theta)( \cos \theta)} \\ \implies \frac{ { \cos}^{2} \theta + 1 + { \sin}^{2} \theta + 2 \sin \theta }{ \cos \theta(1 + \sin \theta)} \\ \implies \frac{2 + 2 \sin \theta}{ \cos \theta(1 + \sin \theta)} \\ \implies \frac{2(1 + \sin \theta)}{ \cos \theta(1 + \sin \theta)}$

now you can cut 1+sinQ and then using identity make LHS=RHS

Let's see how

$we \: know \: \frac{1}{ \cos \theta } = \sec \theta \\ therefore \\ \frac{2}{ \cos \theta } = 2 \sec \theta = LHS$

$\large RHS = 2 \sec \theta$

$\huge LHS = RHS$

Hence proved