Trigonometry Help Please! I've been trying to work it out for hours!?

To Prove : sin x + cos x = sin x / 1- cot x + cos x / 1- tan x

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RHS = [ sin x / ( 1 - cot x ) ] + [ cos x / ( 1 - tan x ) ]

.........= [ s / ( 1 - c/s ) ] + [ c / ( 1 - s/c ) ]

.........= [ s / ( s-c ) / s ] + [ c / ( c-s ) / c ]

.........= [ s² / ( s-c ) ] - [ c² / ( s-c ) ] .............. Note This Step

.........= ( s² - c² ) / ( s - c )

.........= ( s + c )( s - c ) / ( s - c )

.........= sin x + cos x

.........= LHS .......................................... Q.E.D.

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Happy To Help !

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consider RHS

sinx /(1-cotx) = (sinx) / (1 - cosx/sinx) = (sinx)^2 / (sinx -cosx) = - (sinx)^ 2 / (cosx-sinx)

similarly cosx /1-tan x = (cosx)/(1- sinx/cosx) = (cosx)^2 / (cosx-sinx)

RHS = [(cosx) ^2 / (cosx-sinx) ] -[(sinx) ^ 2 / (cosx-sinx)] = [cos ^2 x -sin^2 x] / (cosx-sinx)]

using (a^2-b^2) = (a-b) (a+b)

= [(cosx -sinx) (cosx + sinx)]/ (cosx - sinx) = cosx + sinx

First, get the right side into nothing but sin's and cos's.

Then, you'll probably have to deal with the four-story fraction.

(a/b) / (c/d) which equals (a d) / (b c)

See how far those two steps take you.