# Thread: trigonometric equation

1. ## trigonometric equation

1. Prove that the equation is true.
cos55*+cos65*+cos175*=0
2. Solve the equation:
sin4x+sin2x=sinx

2. Hello, blertta!

Prove: .$\displaystyle \cos55^o+\cos65^o+\cos175^o \:=\:0$
Sum-to-product identity: .$\displaystyle \cos A + \cos B \:=\:2\cdot\cos\left(\frac{A+B}{2}\right)\cos\left (\frac{A-B}{2}\right)$

$\displaystyle \text{We have: }\;\cos55^o + \underbrace{\cos65^o + \cos175^o}$

. . . . .$\displaystyle = \;\cos55^o + \overbrace{2\cdot\cos120^o\cos55^o}$

. . . . .$\displaystyle = \;\cos55^o + 2\left(-\frac{1}{2}\right)\cos55^o$

. . . . .$\displaystyle = \;\cos55^o - \cos55^o$

. . . . .$\displaystyle =\qquad\quad0$

3. $\displaystyle \sin (4x)+\sin (2x)=2\cos x\sin x + 4 \cos^3 x \sin x - 4 \cos x \sin^3 x$.
So if $\displaystyle \sin x\ne 0$ (You should check $\displaystyle \sin x = 0$ as it might be a solution!): $\displaystyle \cos x \left(1 + 2 \cos^2 x - 2 \sin^2 x\right)=0\iff \cos x\left(1+2\cos 2x\right)=0$.
So if $\displaystyle \cos x\ne 0$ (You should check $\displaystyle \cos x = 0$ as well!): $\displaystyle \cos 2x=-\frac 12$.
You can finish it from here. I hope I didn't make any mistakes.

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# sin 55° cos 65°-cos 55° sin 65°

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