Question

solve (cos 3θ)(cos θ) + 1 = (sin 3θ)(sin θ)

Answer 1

We are given the trigonometric expression

$\mleft(\cos 3\theta\mright)\mleft(\cos \theta\mright)+1=\sin 3\theta\mleft(\sin \theta\mright)$

We will follow the steps below

Step 1: subtract (sin 3θ)(sin θ) from both sides

$\mleft(\cos 3\theta\mright)\mleft(\cos \theta\mright)+1-\mleft(\sin 3\theta\mright)\mleft(\sin \theta\mright)=0$

Step 2: we will make use of the relationship to simplify the expression

$\begin{gathered} \text{let} \\ 3\theta=A \\ \theta=B \end{gathered}$

So that we will have

$(\cos A)(\cos B)+1=(\sin A)(\sin B)$

Hence

$(\cos A)(\cos B)-(\sin A)(\sin B)=-1$

Step 3: We will use the identity

$\cos (A+B)=\cos A\cos B-\sin A\sin B$

Then we will have

$\cos (A+B)=-1$

Step 4: We will recall the initial relationship in step 2 so that

$\cos (A+B)=\cos (3\theta+\theta)$

Thus

$\cos (3\theta+\theta)=-1$

Step 5: Simplify the expression in step 4

$\begin{gathered} \cos (4\theta)=-1 \\ 4\theta=\cos ^{-1}(-1) \\ 4\theta=\pi+2\pi\text{n} \end{gathered}$

Simplifying further we have

$\begin{gathered} \theta=\frac{\pi}{4}+\frac{\pi n}{2} \\ \end{gathered}$