tough question

Question

Prove the identity

 

6 cot a

______=   3 sin 2a  

 

1+cot² a

 

 

rewrite this new expression in terms of sin a and cosa a

Answer 1

Let's use a few trigonometric identities to simplify this expression. It's usually a good idea to translate everything into sines and cosines.

 

First, since 1 + cot²(a) = csc²(a), replace 1 + cot²(a) with csc²(a):

 

6cot(a) / csc²(a) = 3sin(2a)

 

Now, replace cot(a) with cos(a)/sin(a) and replace csc²(a) with 1/sin²(a):

 

6sin²(a)cos(a) / sin(a) = 3sin(2a)

6sin(a)cos(a) = 3sin(2a)

 

Finally, replace sin(2a) with 2sin(a)cos(a):

 

6sin(a)cos(a) = 3(2sin(a)cos(a))

6sin(a)cos(a) = 6sin(a)cos(a)

 

We have now proven the identity, and written the expression in terms of sin(a) and cos(a).