show work plz
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Start by finding a common denominator on the right side, so you can add the numerators.
Cos x/(1+Sin x) + Sin x/Cos x = Sec x
[Cos^2 x + (Sin x)(1+Sin x)]/[(1+Sin x)(Cos x)] = Sec x
Multiply out the numerator:
(Cos^2 x + Sin x + Sin^2 x)/[(1+Sin X)(Cos x)]
Substitute in the Pythagorean identity Sin^2 x + Cos^2 x = 1 in the numerator:
(1+Sin x)/[(1+Sin x)(Cos x)]
Simplify, 1+Sin x is in numerator and denominator:
1/Cos x = Sec x
Cos x/(1+Sin x) + Sin x/Cos x = Sec x
[Cos^2 x + (Sin x)(1+Sin x)]/[(1+Sin x)(Cos x)] = Sec x
Multiply out the numerator:
(Cos^2 x + Sin x + Sin^2 x)/[(1+Sin X)(Cos x)]
Substitute in the Pythagorean identity Sin^2 x + Cos^2 x = 1 in the numerator:
(1+Sin x)/[(1+Sin x)(Cos x)]
Simplify, 1+Sin x is in numerator and denominator:
1/Cos x = Sec x