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