Question

solve the equation using sum or difference identities cos3x cosx=sin3x sinx

Answer 1

This looks convenient: cos(3x)cos(x) - sin(3x)sin(x) = 0

Answer 2

but I have to solve for x so I don't know where to go from there

Answer 3

it's cos3 times x and you come up with x2

Answer 4

how did you get that??

Answer 5

\[\cos(3x)\cos(x)-\sin(3x)\sin (x)=0\]The cosine Sum/Difference identity gives us:\[\cos(A \pm B)=\cos(A)\cos(B) \mp \sin(A)\sin(B)\]This means that our A = 3x, and B = x. The Sum/Difference identity then tells us that we can replace our long equation cos(4x). This is because:\[\cos(4x)=\cos(3x+x)=\cos(3x)\cos(x)-\sin(3x)\sin(x)\]Which is exactly what we need. Thus:\[\cos(4x)=0\]Now simply find all values of x where cosine is 0. Since we are not given an interval for x, we can say that cos(4x) = 0 when 4x = pi/2 +pi*k, where k is an integer.\[\cos(4x)=0 \implies 4x = \frac{ \pi }{ 2 }+\pi k \rightarrow x = \frac{ \pi }{ 8 }+\frac{ \pi k }{ 4 }|k \in \mathbb{Z}\]
@starbot