Question

Can someone explain this to me? My paper wants me to find solutions to these problems:
1. cos(2x)+cosx=0
2. 4sin^2(x)tan(x)-3tan(x)=0
3. cos^2(x)-sin^2(x)-3sinx-2=0
4. cos^2(x)-3sin(x)-2=0

Answer 1

well you are being asked to replace the double angle identity with a trig identity that contains a only a single x

so in the 1st case you should know

\[\cos(2x) = \cos^2(x) - \sin^2(x)\]

call this equation 1

then use the identity

\[\sin^2(x) + \cos^2(x) = 1\]

manipulating the identity

\[\sin^2(x) = 1 - \cos^2(x) \]

substitute this into equation 1 gives

\[\cos(2x) = 2\cos^2(x) - 1\]

this can now be substituted into the 1st equation

\[2\cos^2(x) + \cos(x) - 1 = 0\]

now you have a quadratic equation and can solve for x by factoring.

so the questions you have listed require you to use some trig identities and then solve for x.

hope it helps.