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lorraine_09 · Jul 27, 2009, 07:56 AM

establish this identity:

cosx + 1+sinx = 2 sec x
1+sinx cosx

establish this identity:

sec(squared)x - tan(squared)x + tan x = sin x + cos x
sec x

Thank you so much any help would be great!! :)

Unknown008 · Jul 27, 2009, 08:52 AM

1.

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

Make everything a single fraction:

$\frac{cos^2(x)+(1+sin(x))^2}{(1+sin(x))(cos(x))}$

Then simplify:

$\frac{cos^2(x)+1+2sin(x)+sin^2(x)}{(1+sin(x))(cos( x))}$

$\frac{(cos^2(x)+sin^2(x))+1+2sin(x)}{(1+sin(x))(co s(x))}$

$\frac{1+1+2sin(x)}{(1+sin(x))(cos(x))}$

$\frac{2+2sin(x)}{(1+sin(x))(cos(x))}$

$\frac{2(1+1sin(x))}{(1+sin(x))(cos(x))}$

$\frac{2}{(cos(x))}$

$2sec(x)$

Unknown008 · Jul 27, 2009, 08:58 AM

2.

$\frac{sec^2(x) - tan^2(x) + tan (x)}{sec(x)} = sin( x) + cos( x)$

Use: tan^2(x) + 1 = sec^2(x)

$\frac{(tan^2(x) +1) - tan^2(x) + tan (x)}{sec(x)}$

$\frac{tan^2(x) +1 - tan^2(x) + tan (x)}{sec(x)}$

$\frac{(tan^2(x) - tan^2(x))+1 + tan (x)}{sec(x)}$

$\frac{1 + tan (x)}{sec(x)}$

Convert everything into cos and sin.

$\frac{1 + \frac{sin(x)}{cos(x)}}{\frac{1}{cos(x)}}$

$$ 1 + \frac{sin(x)}{cos(x)}$ \times cos(x)$

$cos(x) + sin(x)$

Hope all helped! :)

lorraine_09 · Jul 27, 2009, 09:12 AM

Yes thank you but how do I make each side equal to the other like 1=1 for example?

Unknown008 · Jul 27, 2009, 09:19 AM

in fact, I should have typed :

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

Then continue with the simplifying. You just have to show that they are equal at the end, putting for example 'shown' or 'proved' at the end.

lorraine_09 · Jul 27, 2009, 09:22 AM

OK yes thank you very much again! Have a great day!

Unknown008 · Jul 27, 2009, 09:26 AM

A 'rating' would be nice from you too! :)

lorraine_09 · Jul 27, 2009, 09:45 AM

OK I'm new at this today actually so I'm sorry but I don't know how to rate you! Could you help me out!

Unknown008 · Jul 27, 2009, 11:47 AM

Just click on the 'rate this answer button, and follow what you are asked! :)