<s>[INST] Verify trigonometry equation $\tan A - \csc A \sec A (1-2\cos^2 A)= \cot A$ How would I verify the following trigonometry identity?
$$\tan A - \csc A \sec A (1-2\cos^2 A)= \cot A$$
My work so far is 
$$\frac{\sin A}{\cos A}-\frac{1}{\sin A}\frac{1}{\cos A}(1- \cos^2 A- \cos^2 A)$$
 [/INST] I just want to comment on how you could take your work so far a little further.
$$\frac{\sin A}{\cos A}-\frac{1}{\sin A}\frac{1}{\cos A}(1- \cos^2 A- \cos^2 A)$$
What you need to do is to get a common denominator
$$\frac{\sin A}{\cos A}\cdot\frac{\sin A}{\sin A}-\frac{1}{\sin A\cos A}(1- \cos^2 A- \cos^2 A)$$
$$=\frac{\sin^2 A}{\sin A\cos A}-\frac{1- \cos^2 A- \cos^2 A}{\sin A\cos A}$$
$$=\frac{\sin^2 A}{\sin A\cos A}-\frac{\sin^2 A- \cos^2 A}{\sin A\cos A}$$
$$=\frac{\cos^2 A}{\sin A\cos A}$$
$$=\frac{\cos A}{\sin A}$$
$$=\cot A$$
But in general I do think thatit would be nice to use more weapons than just breaking everything into sines and cosines.
As a starting point we have
$$\cos^2\theta + \sin^2\theta = 1$$
$$\tan^2\theta + 1 = \sec^2\theta$$
$$\cot^2\theta + 1 = \csc^2\theta$$
$$\tan\theta = \frac{\sin\theta}{\cos\theta}, \cot\theta = \frac{\cos\theta}{\sin\theta}$$
$$\csc\theta = \frac{1}{\sin\theta}, \sec\theta = \frac{1}{\cos\theta}, \cot\theta = \frac{1}{\tan\theta}$$
But we could use the above identities to come up with these identities
$$\cot\theta\tan\theta = \sin\theta\csc\theta = \cos\theta\sec\theta = 1$$
$$\tan\theta = \frac{\sec\theta}{\csc\theta}, \cot\theta = \frac{\csc\theta}{\sec\theta}$$
So lets try the same problem again, but this time deploying these new weapons.
$$\tan A - \csc A \sec A (1-2\cos^2 A)$$
$$=\cot A \tan A(\tan A - \csc A \sec A (1-2\cos^2 A))$$
$$=\cot A( \tan^2 A - \frac{\sec A}{\csc A}\csc A \sec A (1-\frac{2}{\sec^2 A})))$$
$$=\cot A( \tan^2 A - \sec^2 A (1-\frac{2}{\sec^2 A})))$$
$$=\cot A( \tan^2 A - \sec^2 A + 2)$$
$$=\cot A( (\sec^2 A - 1) - \sec^2 A + 2)$$
$$=\cot A$$
The point that I'm trying to make is that sometimes it's more convenient to express everything in terms of sins and coss, but at other times it's better to express everything in terms of tans and secs (or cots and cscs). These other identities will make your trigonometric simplifications all the more powerful.
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