csc^2 x/2=2 secx

it says to put all solutions in [0, 360) interval and to give all exact solutions (so they would end in 360k, where k is an interger). I tried putting everything in sine and cosine, but it didn't work out :eek:

Since you and I can't write in the math script that's possible on this board, at least use parenthesis and functions you might use in basic or Excel.

If that's csc(x/2)^2= 2*sec(x)

and it definitely looks like a mess:

Soln: http://www.wolframalpha.com/input/?i=csc%28x%2F2%29^2%3D+2*sec%28x%29

Hm, that's what I get:

cosec^2 (\frac{x}{2}) = 2 sec\ x

Convert all in cos and sin;

\frac{1}{sin^2 (\frac{x}{2})} = \frac{2}{cos\ x}

Cross multiply:

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

Break cos into double angle using cos2A = 1 - 2sin^2A

We get:

1 - 2sin^2(\frac{x}{2})= 2 sin^2 (\frac{x}{2})

Putting all sin on one side, we get:

1= 4 sin^2 (\frac{x}{2})

\frac14= sin^2 (\frac{x}{2})

\sqrt{\frac14} = \frac12= sin (\frac{x}{2})

sin^{-1}(\frac12)= \frac{x}{2}

The angles for x/2 are 30 and 150 degrees. Then, for x become 60 and 300 degrees.

I hope it helped! :)

Here is another way for kicks.

csc^{2}(\frac{x}{2})=\frac{2}{1-cos(x)}

Rewrite as

\frac{2}{1-cos(x)}=\frac{2}{cos(x)}

2cos(x)=2-2cos(x)

cos(x)=\frac{1}{2}

x=\frac{\pi}{3}, \;\ \frac{5\pi}{3}