Question

Which double angle or half angle identity would you use to verify the following: csc x sec x = 2 csc 2x

Answer 1

b

Explanation:

I would use b.

Why?

`2 \csc(2x)`

`2 (1)/(\sin(2x))`

`(2)/(\sin(2x))`

`(2)/(2\sin(x)\cos(x))`

`\csc(x) \sec(x)`

I applied the identity sin(2x)=2sin(x)cos(x) in line 3 to 4.

Answer 2

Answer: OPTION B.

Explanation:

It is important to remember these identities:

`csc(x)=(1)/(sin(x))\\\\sec(x)=(1)/(cos(x))`

Knowing this, we can say that:

`csc(x) sec(x)=(1)/(sin(x))*(1)/(cos(x))=(1)/(sin(x)*cos(x))`

Now we need to use the following Double angle identity :

`sin(2x)=2sin(x)cos(x)`

And solve for
`sin(x)cos(x)`:

`(sin(2x))/(2)=sin(x)cos(x)`

The next step is to make the substitution into
`(1)/(sin(x)*cos(x))` and finally simplify:

`(1)/((sin(2x))/(2))=((1)/(1))/((sin(2x))/(2))=(2)/(sin(2x))=2csc(2x)`