Question

Verify the identity.

cotangent of x divided by quantity one plus cosecant of x equals quantity cosecant of x minus one divided by cotangent of x
Please! Thank you!

Answer 1

(cosx/ sin x)/ 1+ 1/sin x
(cos x/ sin x) * (cos x * sin x/ sin x^2)
(cos x * sin x) + (cos x * sin x) /sin^2 x

That is as far as I got, but I don't think it is right.

Answer 2

Well that was getting me nowhere (changing to sin and cos) so I'll just cross multiply and go from there

\[\large \frac{cot(x)}{1 + csc(x)} = \frac{csc(x) - 1}{cot(x)}\]

Cross multiply

\[\large cot^2(x) = (1 + csc(x))(csc(x) - 1)\]

Now change everything to sin and cos

\[\large (\frac{cos(x)}{sin(x)})^2 = (1 + \frac{1}{sin(x)})(\frac{1}{sin(x)} - 1)\]
Put the fractions over common denominators of \(\large sin\)
\[\large (\frac{cos(x)}{sin(x)})^2 = (\frac{1 + sin(x)}{sin(x)})(\frac{1 - sin(x)}{sin(x)})\]
Multiply that out
\[\large \frac{cos^2(x)}{sin^2(x)} = \frac{(1 + sin(x))(1 - sin(x))}{sin^2(x)}\]
\[\large \frac{cos^2(x)}{\cancel{sin^2(x)}} = \frac{(1 + sin(x))(1 - sin(x))}{\cancel{sin^2(x)}}\]

\[\large cos^2(x) = (1 + sin(x))(1 - sin(x))\]
\[\large cos^2(x) = 1 - sin^2(x)\]
\[\large cos^2(x) = cos^2(x)\]

Woo! :) sorry lot of typing

Answer 3

Thank you so much!!!

Answer 4

Anytime :)