Explanation:
14. You can multiply the first fraction by a suitable form of 1, then simplify.
[tex]\dfrac{\cos{x}}{1+\sin{x}}+\dfrac{1+\sin{x}}{\cos{x}}=\dfrac{\cos{x}}{1+\sin{x}}\cdot\dfrac{1-\sin{x}}{1-\sin{x}}+\dfrac{1+\sin{x}}{\cos{x}}\\\\=\dfrac{\cos{x}(1-\sin{x})}{1-\sin^2{x}}+\dfrac{1+\sin{x}}{\cos{x}}=\dfrac{\cos{x}}{\cos{x}}\cdot\dfrac{1-\sin{x}}{\cos{x}}+\dfrac{1+\sin{x}}{\cos{x}}\\\\=\dfrac{1-\sin{x}+1+\sin{x}}{\cos{x}}=\dfrac{2}{\cos{x}}=2\sec{x}[/tex]
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15. This makes use of a rather obscure cotangent identity.
[tex]\displaystyle\cot{\left(x-\frac{\pi}{2}\right)}=\frac{\cot{x}\cot{\frac{\pi}{2}}+1}{\cot{\frac{\pi}{2}}-\cot{x}}\\\\=\frac{0+1}{0-\cot{x}}=\frac{-1}{\cot{x}}=-\tan{x}[/tex]
