Question

Tan^2a-cot^2a = sec^2a (1-cot^2a) prove​

Answer 1

See explanation

Explanation:

Use the definitions:

`\tan \alpha=(\sin \alpha)/(\cos \alpha)\\ \\\cot \alpha=(\cos \alpha)/(\sin \alpha)\\ \\\sec \alpha=(1)/(\cos \alpha)\\ \\\csc \alpha=(1)/(\sin \alpha)\\ \\`

Now,

`\tan^2\alpha -\cot^2\alpha=(\sin^2\alpha)/(\cos^2\alpha)-(\cos^2\alpha)/(\sin^2\alpha)=(\sin^4\alpha-\cos ^4\alpha)/(\sin^2\alpha\cos ^2\alpha )=\\ \\=((\sin^2\alpha-\cos ^2\alpha)(\sin^2\alpha-\cos ^2\alpha))/(\sin^2\alpha\cos ^2\alpha )=((\sin^2\alpha-\cos ^2\alpha)\cdot 1)/(\sin^2\alpha\cos ^2\alpha )`

and

`\sec^2\alpha(1-\cot^2\alpha)=(1)/(\cos^2 \alpha)\left(1-(\cos^2\alpha)/(\sin^2\alpha)\right)=(1)/(\cos^2 \alpha)\left((\sin^2\alpha-\cos^2\alpha)/(\sin^2\alpha)\right)=\\ \\=(\sin^2\alpha-\cos ^2\alpha)/(\sin^2\alpha\cos ^2\alpha)`

As you can see, left and right parts simplify to the same expression, so left and right parts are the same.