Question

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

Answer 1

\(\Large \frac ab + \frac cd = \frac {ad+bc}{bd}\).
Simplify.

Answer 2

how would that look? @aum

Answer 3

Its a very long process, but I can show you...it will just take quite a bit of time. It's up to you if you want to see the whole thing.

Answer 4

I think I figured it out. would it look like this?

Answer 5

(1+sin x)/(cos x)+(cos x)/(1+sin x)

(1+sin x)/(cos x)+[(cos x)(1-sin x)]/[(1+sin x)(1-sin x)]

(1+sin x)/(cos x)+[(cos x)(1-sin x)]/(1-sin²x)

(1+sin x)/(cos x)+[(cos x)(1-sin x)]/(cos²x)

(1+sin x)/(cos x)+(1-sin x)/(cos x)

1/cos x+tan x+1/cos x-tan x

sec x+sec x

2secx

Answer 6

Mine looks nothing like that at all. But if you know for a fact that you used your identities properly, then you should be good. It's the use of the identities that makes it either correct or incorrect.

Answer 7

can you show me how you would do it? It might be easier for next time

Answer 8

\(\Large
\frac {\cos x}{1+\sin x}+\frac{1+\sin x}{\cos x} =
\frac{ (\cos x)(\cos x) + (1+\sin x)(1+\sin x) }{(1+\sin x)(\cos x)} = \\ \Large
\frac {\cos^2 x + 1 + 2\sin x + \sin^2 x}{\cos x(1+\sin x)} = \\ \Large
\frac {2 + 2\sin x }{\cos x(1+\sin x)} = \frac {2(1+\sin x) }{\cos x(1+\sin x)} =
\frac{2}{\cos x} = 2\sec x
\)

Answer 9

Thank you :) @aum

Answer 10

You are welcome.