Question

help verifying this equation. just need the steps.

(sin x - cos x/sin x) + (cos x - sin x/ cos x) = 2 - (sec x)(csc x)

Answer 1

From LHS:
(sin x - cos x/sin x) + (cos x - sin x/ cos x) =
\[[(\sin ^{2}x \cos x - \cos ^{2} x) + (\cos ^{2}x \sin x - \sin ^{2} x)] / (\sin x \cos x)\] =
\[[ \sin ^{2}x \cos x + \cos ^{2}x \sin x - (\sin ^{2}x + \cos ^{2}x) ] / (\sin x \cos x)\] =
\[(\sin x \cos x + \sin x \cos x -1) / (\sin x \cos x)\] =
\[(2\sin x \cos x -1) / (\sin x \cos x)\] =
\[2 - (1 / \sin x \cos x)\] = 2 - sec(x)cosec(x) = RHS
(Proven)

Answer 2

Hi,

Sorry, I made an error in the fourth line of my proof for this question, but then again, this equation has an error - are you certain that you typed it out correctly? There is no way that sin x + cos x =2 (if you see my working above) which would be required to satisfy this proof.

Answer 3

Hi, yes i checked the question twice and I typed it out correctly. I figured the problem out but thx anyways. My teacher doesn't want us to write the proofs so we don't write them or identify them. Plus i think you made an error or went another route to solve this problem from the beginning. Because when I multiplied the denominators together I got a different product than you.

Answer 4

[ (cosx(sinx - cosx) + sinx(cosx - \sin x) ] / cosxsinx

Answer 5

the answer would be [ coxsinx - cos^2x + sinxcosx - sin^2x ] / cosxsinx = 2 - (sescx)(cscx)

Answer 6

I think you made an error in your multiplication but thanks anyways for trying to help me :)

Answer 7

@aminehr : Looking back at your solution and the question, I think I know where the source of the error lies - confusion was created by your typing of the parentheses (& the spacing) in your original question. Your original question was phrased as follows:

(sin x - cos x/sin x) + (cos x - sin x/ cos x) = 2 - (sec x)(csc x)

This gave me the impression that the LHS would read:
(sin x - cos x/sin x) + (cos x - sin x/ cos x) = (sin x - cot x) + (cos x - tan x)

while what you actually intended to write was:
[(sin x - cos x)/sin x] + [(cos x - sin x)/ cos x] = 2 - (sec x)(csc x)

Thanks.