Question 252449
My general style is to turn things into sin and cos. I tell kids, we understand those better than the others. What have to do first is decide which side looks more complicated and turn it into the less complicated side. I will try to turn the left into the right. So,
(tan A + Cot A)^2 = Sec^ 2 A + Cosec^2 A
becomes
(sin A / cos A + cos A / Sin A)^2

Adding these fractions with common denominators gets us
(sin^2 A + cos^2 A) / (cos^2 A * Sin^2 A)

We have an identity in the numerator, so we now get

1 / (Cos^2 A * Sin^2 A)

This can be expressed as

1/cos^2 A + 1/sin^2 A

which then becomes

sec^2 A + cosec^2 A. This is the right hand side, so we are done.