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Explanation:
For proofs of this nature, it often works well to replace everything possible with appropriate equivalents. Here, we'll use sin² to represent sin²(A), and equivalent notation for other trig functions.
We will change the left side to the form of the right side.
[tex]\displaystyle \frac{\tan^2-\cot^2}{1+\cot^2}=\frac{\sin^2-\cos^2}{\cos^2}\\\\=\frac{(\sec^2-1)-(\csc^2-1)}{\csc^2}\\\\=(\sin^2)(\sec^2-\csc^2)\\\\=\frac{\sin^2}{\cos^2}-1=\frac{\sin^2-\cos^2}{\cos^2}\qquad\text{Q.E.D.}[/tex]
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The equivalents (identities) used are ...
1+tan² = sec²
1+cot² = csc²
1/csc = sin
sec = 1/cos
