Answer:
[tex]x=18\textdegree[/tex]
Step-by-step explanation:
We are given the equation:
[tex]\sin(2x)\sin(3x)=\cos(2x)\cos(3x)[/tex]
And we want to find the smallest positive value of x such that it makes the equation true.
We can rewrite our equations as:
[tex]\displaystyle \cos(2x)\cos(3x) - \sin(2x)\sin(3x) = 0[/tex]
Recall that:
[tex]\displaystyle \cos (\alpha + \beta) = \cos \alpha\cos-\sin\alpha\sin\beta[/tex]
Therefore, by letting α = 2x and β = 3x, we obtain:
[tex]\displaystyle \begin{aligned} \cos(2x)\cos(3x) - \sin(2x)\sin(3x) & = \cos (2x + 3x) \\ \\ &= \cos 5x \end{aligned}[/tex]
Hence:
[tex]\displaystyle \cos 5x = 0[/tex]
Recall the Unit Circle. The smallest angle value for which cosine equals 0 is cos(90°). Thus:
[tex]\displaystyle 5x = 90^\circ[/tex]
Therefore:
[tex]\displaystyle x = 18^\circ[/tex]
In conclusion:
[tex]\displaystyle x = 18^\circ[/tex]
