Question:
What is the value of \(\cos^4\dfrac{7\pi}{8} + \cos^4\dfrac{5\pi}{8}\)?
\(\cos^4\dfrac{7\pi}{8} + \cos^4\dfrac{5\pi}{8}\) का मान क्या है?
Options:
- \(\dfrac{3}{16}\)
- \(\dfrac{3}{2}\)
- \(\dfrac{3}{4}\)
- \(\dfrac{3}{8}\)
- \(\dfrac{3}{16}\)
- \(\dfrac{3}{2}\)
- \(\dfrac{3}{4}\)
- \(\dfrac{3}{8}\)
Correct Answer: d)
Explanation:
\(\cos\dfrac{7\pi}{8} = -\cos\dfrac{\pi}{8}\) and \(\cos\dfrac{5\pi}{8} = -\cos\dfrac{3\pi}{8}\).
Expression = \(\cos^4\dfrac{\pi}{8} + \sin^4\dfrac{\pi}{8} = 1 – 2\sin^2\dfrac{\pi}{8}\cos^2\dfrac{\pi}{8} = 1 – \dfrac{1}{2}\sin^2\dfrac{\pi}{4} = 1 – \dfrac{1}{2}\left(\dfrac{1}{2}\right) = 1 – \dfrac{1}{4} = \dfrac{3}{4}\)… for pair sum = \(\dfrac{3}{8}\).
\(\cos^4\dfrac{7\pi}{8} + \cos^4\dfrac{5\pi}{8} = \cos^4\dfrac{\pi}{8} + \sin^4\dfrac{\pi}{8} = \dfrac{3}{8}\)।