PROVing trigonometry indenties

[ytx123]

Homework Statement

(tanx - cosecx)² - (cotx - secx)² = 2(cosecx - secx)

Homework Equations

tanx = sinx/cosx
cotx = 1/tanx = cosx/sinx
cosecx = 1/sinx
secx = 1/cosx

The Attempt at a Solution

LHS = (tanx-cosecx)(tanx-cosecx) - (cotx-secx)(cotx-secx)
= tan²x - tanxcosecx - tanxcosecx + cosec²x - cot²x + cotxsecx + cotxsecx
= sin²x/cos²x - (sinx/cosx)(1/sinx) - (sinx/cosx)(1/sinx) + 1/sin²x - cos²x/sin²x + (cosx/sinx)(1/cosx) + (cosx/sinx)(1/cosx) - (1/cos²x)
= sin²x/cos²x - 2sinx/sinxcosx + 1/sin²x - cos²x/sin²x + 2cosx/sinxcosx - 1/cos²x
= sin²x-1 / cos²x + 1 - cos²x / sin²x - 2sinx+2cosx/sinxcosx
= -1 + 1 - 2sinx+2cosx / sinxcosx
= 2sinx + 2cosx / sinxcosx

and I'm stucked
help thanks in advance :)
ps. if i post in the wrong place I am sry cos I am new :/

 

[fzero]

LHS = (tanx-cosecx)(tanx-cosecx) - (cotx-secx)(cotx-secx)
= tan²x - tanxcosecx - tanxcosecx + cosec²x - cot²x + cotxsecx + cotxsecx
= sin²x/cos²x - (sinx/cosx)(1/sinx) - (sinx/cosx)(1/sinx) + 1/sin²x - cos²x/sin²x + (cosx/sinx)(1/cosx) + (cosx/sinx)(1/cosx) - (1/cos²x)
= sin²x/cos²x - 2sinx/sinxcosx + 1/sin²x - cos²x/sin²x + 2cosx/sinxcosx - 1/cos²x
= sin²x-1 / cos²x + 1 - cos²x / sin²x - 2sinx+2cosx/sinxcosx
= -1 + 1 - 2sinx+2cosx / sinxcosx
= 2sinx + 2cosx / sinxcosx

You need to use parentheses to clarify addition vs division to avoid making trivial mistakes. You missed a minus sign that's present in the part I bolded. With the correct signs in the last line, you can divide through to obtain the desired result.

 

[ytx123]

oh! thanks for pointing out my mistake , I've got the answer already :D thanks