Answer:
[tex]\theta = \dfrac{1}{2}\pi k[/tex]
[tex]\theta = \dfrac{\pi}{4}+\dfrac{1}{2}\pi k[/tex]
Step-by-step explanation:
We are solving for θ in the trigonometric equation:
[tex](\sin(\theta))^8 -(\cos(\theta))^8 = (\sin(\theta))^2 -(\cos(\theta))^2[/tex]
We can rewrite the right side using the double angle identity for cosine:
- [tex]\cos(2\theta)=(\cos(\theta))^2-(\sin(\theta))^2[/tex]
- [tex]a-b = -(b-a)[/tex]
↓↓↓
[tex](\sin(\theta))^8 -(\cos(\theta))^8 = -\cos(2\theta)[/tex]
Beyond that, there's nothing we can do to simplify the equation further, so we must resort to graphing. See the attached image.
We can see that the two expressions are equal at their amplitudes and x-intercepts, which are:
[tex]\boxed{\dfrac{1}{2}\pi k}[/tex]
[tex]\boxed{\dfrac{\pi}{4}+\dfrac{1}{2}\pi k}[/tex]
