Prove that

$(\sin A+cosec A{)}^2+(\cos A+\sec A{)}^2=7+{\tan }^{2}A+{\cot }^{2}A$.

Simplify

$(1-\cos \theta )(1+\cos \theta )\left(1+{\cot }^2\theta \right)$

Prove that $\left(\frac{1+{\tan }^{2}A}{1+{\cot }^2\mathrm{A}}\right)={\left(\frac{1+\tan A}{1-\cot A}\right)}^2={\tan }^{2}A$.

*NOTE: THE SECOND PART OF THE ABOVE GIVEN QUESTION IN THE BOOK IS WRONG, THE CORRECT QUESTION IS WRITTEN BELOW*

$\left(\frac{1+{\tan }^{2}A}{1+{\cot }^2\mathrm{A}}\right)={\left(\frac{1+\tan A}{1+\cot A}\right)}^2={\tan }^{2}A$