Question

Verify the identity
(cos x)/(1+sin x) + (1+sin x)/(cos x) = 2sec x

Answer 1

\[(\cos x)/(1+\sin x) \times (\cos x)/(\cos x)+ (1+\sin x)/(\cos x) \times (1+\sin x)/(1+sinx)\]

Answer 2

\[\cos x + 1 + \sin x\]

Answer 3

How do I go from there?

Answer 4

hehe

Answer 5

$$\bf \cfrac{cos(x)}{1+sin(x)}+\cfrac{1+sin(x)}{cos(x)} \implies
\cfrac{cos^2(x)+(1+sin^2(x))}{(1+sin(x))cos(x)}\\
\cfrac{cos^2(x)+1+2sin^2(x)+sin^2(x)}{(1+sin(x))cos(x)}\\
\color{blue}{cos^2(x)+sin^2(x)=1}\\
\cfrac{cos^2(x)+1+2sin(x)+sin^2(x)}{(1+sin(x))cos(x)} \implies
\cfrac{1+1+2sin(x)}{(1+sin(x))cos(x)}\\
\cfrac{2+2sin(x)}{(1+sin(x))cos(x)} \implies
\cfrac{2\cancel{(1+sin(x))}}{(\cancel{1+sin(x))}cos(x)}
$$