Question #71ebe

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Answer 1 · 2017-04-07T15:51:11

I have changed the question so that it is an identity. (Without the parentheses it is not.)

Start with

$(1+cosx)/(1-cosx) - (1-cosx)/(1+cosx)$

We want a single quotient, so get a common denominator and simplify.

$(1+cosx)/(1-cosx) - (1-cosx)/(1+cosx) = ((1+cosx)(1+cosx) - (1-cosx)(1-cosx))/((1-cosx)(1+cosx))$

$= ((1+2cosx+cos^2x)-(1-2cosx+cos^2x))/(1-cos^2x)$

It's hard to see whether this helped, so keep simplifying.

$= (1+2cosx+cos^2x-1+2cosx-cos^2x)/(1-cos^2x)$

$= (4cosx)/(1-cos^2x)$

Now, at least I see a $4$ in the numerator, so that's good. We want a $sinx$ in the denominator, so use the Pythagorean identity to get

$= (4cosx)/sin^2x$

Now, recall that $tanx = sinx/cosx$ and rewrite

$= (4cosx)/(sinxsinx) = 4/(sinxsinx/cosx) = 4/(sinxtanx)$