tan A + cot A = 2csc(2A)

Question

I am so lost

Answer 1

This looks like an identity you have to prove, right?

 

Start with the left side and write it in terms of sin and cos:

 

tan A + cot A = sin A / cos A + cos A / sin A    

= (sin² A + cos² A) / (sin A cos A)       (By adding the fractions)

= 1 / (sin A cos A)                             (Because of the Pythagorean identity, sin² A + cos² A = 1

 

Now what about the right side:

 

2csc(2A) = 2 / sin (2A)        (Since sin and csc are reciprocals of each other)

= 2 / (2 sin A cos A)            (By the double-angle formula for sin)

= 1 / (sin A cos A)               (Cancel out the 2 in the numerator and denominator)

 

So the left side and the right side are both equal to 1 / (sin A cos A), so they are equal to each other.