The expression that is not equivalent to [tex]\cos (\dfrac{\pi}{5})[/tex] is:
3. [tex]\cos(\dfrac{4\pi}{5})[/tex]
We are asked to find which of the expression is not equivalent to:
[tex]\cos (\dfrac{\pi}{5})[/tex]
1)
[tex]\cos(\dfrac{-\pi}{5})[/tex]
We know that:
[tex]\cos(-\theta)=\cos(\theta)[/tex]
Hence, we get:
[tex]\cos(\dfrac{-\pi}{5})=\cos(\dfrac{\pi}{5})[/tex]
Hence, option: 1 is incorrect.
2)
[tex]\cos(\dfrac{9\pi}{5})[/tex]
We know that:
[tex]\cos(\dfrac{9\pi}{5})=\cos( 2\pi-\dfrac{\pi}{5})[/tex]
As we know that:
[tex]\cos(2\pi-\theta)=\cos(\theta)[/tex]
Hence, we have:
[tex]\cos(\dfrac{9\pi}{5})=\cos(\dfrac{\pi}{5})[/tex]
Hence, option: 2 is incorrect.
4)
[tex]\cos(\dfrac{11\pi}{5})[/tex]
We know that:
[tex]\cos(\dfrac{11\pi}{5})=\cos( 2\pi+\dfrac{\pi}{5})[/tex]
As we know that:
[tex]\cos(2\pi+\theta)=\cos(\theta)[/tex]
Hence, we have:
[tex]\cos(\dfrac{11\pi}{5})=\cos(\dfrac{\pi}{5})[/tex]
Hence, option: 4 is incorrect.
3)
[tex]\cos(\dfrac{4\pi}{5})[/tex]
We know that:
[tex]\cos(\dfrac{4\pi}{5})=\cos(\pi-\dfrac{\pi}{5})[/tex]
As we know that:
[tex]\cos(\pi-\theta)=-\cos(\theta)[/tex]
Hence, we have:
[tex]\cos(\dfrac{4\pi}{5})=-\cos(\dfrac{\pi}{5})\neq \cos (\dfrac{\pi}{5})[/tex]
Hence, option: 3 is the answer.
