Question

व्यंजक ${\displaystyle {\cos }^4 \frac{\pi }{8}+{\cos }^4 \frac{3\pi }{8}+{\cos }^4 \frac{5\pi }{8}+{\cos }^4 \frac{7\pi }{8}}$ का मान ज्ञात कीजिए।

[संकेत: व्यंजक ${\displaystyle 2({\cos }^4 \frac{\pi }{8}+{\cos }^4 \frac{3\pi }{8})=2[{({\cos }^2 \frac{\pi }{8}+{\cos }^2 \frac{3\pi }{8})}^2-2{\cos }^2 \frac{\pi }{8}{\cos }^2 \frac{3\pi }{8}]}$ के रूप में सरल कीजिए।

Answer 1

= ${\displaystyle {\cos }^4 \frac{\pi }{8}+{\cos }^4 \frac{3\pi }{8}+{\cos }^4 \frac{5\pi }{8}+{\cos }^4 \frac{7\pi }{8}}$ की गणना करें।

अभिव्यक्ती को हल करने पर,

= ${\displaystyle {\cos }^4 \frac{\pi }{8}+{\cos }^4 \frac{3\pi }{8}+{\cos }^4(\pi -\frac{3\pi }{8})+{\cos }^4(\pi -\frac{\pi }{8})}$

= ${\displaystyle {\cos }^4 \frac{\pi }{8}+{\cos }^4 \frac{3\pi }{8}+{\cos }^4 \frac{3\pi }{8}+{\cos }^4 \frac{\pi }{8}}$

= ${\displaystyle 2[{\cos }^4 \frac{\pi }{8}+{\cos }^4 \frac{3\pi }{8}]}$

हल करने पर,

= ${\displaystyle 2[{\cos }^4 \frac{\pi }{8}+{\cos }^4 (\frac{\pi }{2}-\frac{\pi }{8})]}$

= ${\displaystyle 2[{\cos }^4 \frac{\pi }{8}+{\sin }^4 \frac{\pi }{8}]}$

दिये हुए संकेत का प्रयोग करके हल करने पर,

${\displaystyle 2[{\cos }^4 \frac{\pi }{8}+{\sin }^4 \frac{3\pi }{8}]=2[{({\cos }^2 \frac{\pi }{8}+{\sin }^2 \frac{3\pi }{8})}^2-2{\cos }^2 \frac{\pi }{8}{\sin }^2 \frac{3\pi }{8}]}$

= ${\displaystyle 2[1 -2{\cos }^2 \frac{\pi }{8}{\sin }^2 \frac{3\pi }{8}]}$

= ${\displaystyle [2-4{\cos }^2 \frac{\pi }{8}{\sin }^2 \frac{3\pi }{8}]}$

= ${\displaystyle [2- 4{\cos }^2 \frac{\pi }{8}{\sin }^2 \frac{3\pi }{8}]}$

= ${\displaystyle [2-{\left(2\cos \frac{\pi }{8}\sin \frac{3\pi }{8}\right)}^2]}$

2sinθ cosθ = sin2θ में लागू करने पर,

= ${\displaystyle 2-{\left(\sin \frac{\pi }{4}\right)}^2}$

= ${\displaystyle 2-{\left(\frac{1}{\sqrt{2}}\right)}^2}$

= ${\displaystyle 2-\frac{1}{2}}$

= ${\displaystyle \frac{3}{2}}$

दी गई अभिव्यक्ती का मान ${\displaystyle \frac{3}{2}}$ है।