Establish each identity below.
(1) 9 sec^2 (x) - 5 tan^2 (x) = 5 + 4 sec^2 (x)
(2) cos(x)/[1 + sin(x)] + [1 + sin(x)]/cos(x) = 2 sec (x)
#1:
Note that $\displaystyle \tan^{2} x = \sec^{2} x - 1$. So substitute it in and simplify.
#2:
Combine into a single fraction:
$\displaystyle \frac{\cos x}{1+\sin x} {\color{blue}\cdot \frac{\cos x}{\cos x} } \: + \: \frac{1+ \sin x}{\cos x} {\color{blue} \cdot \frac{1+ \sin x}{1 + \sin x} }$
$\displaystyle = \frac{\cos^2 x + (1+ \sin x)^2}{\cos x(1+ \sin x)}$
Expand the numerator and simplify.
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