Verifying two trigonometric identities with cotangent and sine

[andyc18]

I need hep verifying two trig identity problems

1. (cot-csc)^2 = (1-cos)/(1+cos)


2. sin2tan2/tan2-sin2 = 1

Any help?
Thanks in advance!

 

[Dr Transport]

The easiest way to prove trig identities is to express all of the quantites in terms of sines and cosines, then start rearranging from there...

 

[andyc18]

I have tried doing that many times ..still having no luck..

 

[James R]

Could you please write out your questions again, with the variables, and all bracketing correct?

There's no way to tell is, by sin2 you mean (sin x)² or sin 2x.

Also, in your second problem, do you mean

$$\frac{\sin^2 x \tan^2 x}{\tan^2 x} - \sin^2 x$$

or something else?

Notation is important.

 

[James R]

Here's a solution to the first problem:

$$(\cot x - \mbox{cosec} x)^2\\<br /> = \left(\frac{\cos x}{\sin x} - \frac{1}{\sin x}\right)^2\\<br /> = \frac{(\cos x - 1)(\cos x - 1)}{\sin^2 x}\\<br /> = \frac{(\cos x - 1)(\cos x - 1)}{1 - \cos^2 x}\\<br /> = \frac{(\cos x - 1)(\cos x - 1)}{(1 + \cos x)(1 - \cos x)}\\<br /> = \frac{1 - \cos x}{1 + \cos x}$$