Question

prove that pls tomorrow exam​

Answer 1

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Answer 2

Step-by-step explanation:

1.

$\frac{ \cos(x) }{1 + \sin(x) } \times \frac{1 - \sin(x) }{1 - \sin(x) } + \frac{1 + \sin(x) }{ \cos(x) } \\ \\ = \frac{ \cos(x) (1 - \sin(x)) }{1 {}^{2} - \sin {}^{2} (x) } + \frac{1 + \sin(x) }{ \cos(x) } \\ \\ = \frac{ \cancel { \cos(x)} (1 - \sin(x)) }{\cos {}^{ \cancel2} (x) } + \frac{1}{ \cos(x) } + \frac{ \sin(x) }{ \cos(x) } \\ = \frac{1}{ \cos(x) } - \frac{ \sin(x) }{ \cos(x) } + \frac{1}{ \cos(x) } + \frac{ \sin(x) }{ \cos(x) } \\ = \frac{2}{ \cos(x) } \\ \\ = 2 \sec(x)$

2.

$\sqrt{ \frac{1 - \cos(x) }{1 + \cos(x) } \times \frac{1 - \cos(x) }{1 - \cos(x) } } \\ \\ = \sqrt{ \frac{ (1 - \cos(x) ) {}^{2} }{1 - \cos {}^{2} (x) }} \\ \\ = \frac{1 - \cos(x) } { \sqrt{\sin {}^{2} (x) }} \\ \\ = \frac{1 - \cos(x) }{ \sin(x) } \\ = \csc(x) - \cot(x)$