Question

Solve this question :​

Answer 1

9514 1404 393

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.

  $\displaystyle \frac{\tan^2-\cot^2}{1+\cot^2}=\frac{\sin^2-\cos^2}{\cos^2}\\\\=\frac{(\sec^2-1)-(\csc^2-1)}{\csc^2}\\\\=(\sin^2)(\sec^2-\csc^2)\\\\=\frac{\sin^2}{\cos^2}-1=\frac{\sin^2-\cos^2}{\cos^2}\qquad\text{Q.E.D.}$

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The equivalents (identities) used are ...

  1+tan² = sec²

  1+cot² = csc²

  1/csc = sin

  sec = 1/cos