Prove cosx/(1-tanx) - cosx = sin x - sinx/(1-cotx)

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cosx/(1-tanx) - cosx = sinx-sinx/(1-cotx)

Let us group similar terms:

==> cosx/(1-tanx) + sinx/(a-cotx) = sinx + cosx

We know that :

tanx = sinx/cosx

cotx = cosx/sinx

==> cosx/(1- sinx/cosx) + sinx/(1- cosx/sinx) = sinx + cosx

Now let us simplify:

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

==> cos^2 x/(cosx-sinx  +  sin^2 x/(sinx-cosx) = sinx+cosx

Let us rewrite" sin^2x/(sinx-cosx) = -sin^2 x/(cosx-sinx)

--> (cos^2 x - sin^2 x)/(cosx-sinx) = sinx+cosx

We know that:

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

==> (cosx-sinx)(cosx + sinx)/(cosx-sinx) = sinx+cosx

Reduce similar terms:

==> cosx + sinx = sinx + cosx

Then the equality is true.

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