Question

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

Answer 1

Answer:

b

Step-by-step explanation:

I would use b.

Why?

$2 \csc(2x)$

$2 \frac{1}{\sin(2x)}$

$\frac{2}{\sin(2x)}$

$\frac{2}{2\sin(x)\cos(x)}$

$\frac{1}{\sin(x)\cos(x)}{/tex][tex]\frac{1}{\sin(x)\frac{1}{\cos(x)}$

$\csc(x) \sec(x)$

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

Answer 2

Answer: OPTION B.

Step-by-step explanation:

It is important to remember these identities:

$csc(x)=\frac{1}{sin(x)}\\\\sec(x)=\frac{1}{cos(x)}$

Knowing this, we can say that:

$csc(x) sec(x)=\frac{1}{sin(x)}*\frac{1}{cos(x)}=\frac{1}{sin(x)*cos(x)}$

Now we need to use the following Double angle identity :

$sin(2x)=2sin(x)cos(x)$

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

$\frac{sin(2x)}{2}=sin(x)cos(x)$

The next step is to make the substitution into $\frac{1}{sin(x)*cos(x)}$ and finally simplify:

$\frac{1}{\frac{sin(2x)}{2}}=\frac{\frac{1}{1}}{\frac{sin(2x)}{2}}=\frac{2}{sin(2x)}=2csc(2x)$