Find the value of :-
[tex]\bf \cos^4\dfrac{\pi}{8} +\cos^4\dfrac{3\pi}{8} + \cos^4\dfrac{5\pi}{8} + \cos^4 \dfrac{7\pi}{8}[/tex]
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Find the value of :-
[tex]\bf \cos^4\dfrac{\pi}{8} +\cos^4\dfrac{3\pi}{8} + \cos^4\dfrac{5\pi}{8} + \cos^4 \dfrac{7\pi}{8}[/tex]
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[tex]\bf\huge{\pink{Question}}[/tex]
[tex] \\ [/tex]
What is the value of cos^4 (pi/8) +cos^4 (3pi/8) +cos^4 (5pi/8) +cos^4 (7pi/8)?
[tex]\bf\huge{\pink{Answer}}[/tex]
[tex] \\ [/tex]
cos^4(pi/8)=cos^4(pi-pi/8)=cos^4(7pi/8)—————-(1)
cos^4(3pi/8)=cos^4(pi-3pi/8)=cos^4(5pi/8)————-(2)
therefore equation becomes;
2cos^4(pi/8)+2cos^4(3pi/8)—————(3)
we know cos(x)^2=1/2*(1+cos(2x))————-(4)
and cos(x)^4=(cos(x)^2)^2——————-(5)
therefore equation (3) becomes
2*((cos(pi/8)^2)^2+(cos(3pi/8)^2)^2)————-(6)
applying (4) in (6) we have
2*((1/2*(1+cos(2*pi/8))^2)+(1/2*(1+cos(2*3pi/8))^2))
2*1/4*((1+cos(pi/4))^2+(1+cos(3pi/4))^2))———(7)
cos(pi/4)=1/sqrt(2);cos(3pi/4)=-1/sqrt(2)
therefore (7)=1/2*((1+1/sqrt(2))^2+(1-/sqrt(2))^2)
1/2*((1+1/2+1/sqrt(2))+(1+1/2–1/sqrt(2)))
1/2*(3)
=3/2
Verified answer
Answer :-
Solution :-
[tex]\sf : \; \implies cos^{4} \dfrac{\pi}{8} + cos^{4} \dfrac{3 \pi}{8} + cos^{4} \dfrac{5 \pi}{8} + cos^{4} \dfrac{ 7 \pi}{8}[/tex]
[tex]\sf : \; \implies cos^{4} \dfrac{\pi}{8} + cos^{4} \dfrac{3 \pi}{8} + cos^{4} \bigg( \pi - \dfrac{3\pi}{8} \bigg) + cos^{4} \bigg( \pi - \dfrac{ \pi}{8} \bigg)[/tex]
[tex]\sf : \; \implies cos^{4} \dfrac{\pi}{8} + cos^{4} \dfrac{3 \pi}{8} + cos^{4} \dfrac{3 \pi}{8} + cos^{4} \dfrac{ \pi}{8}[/tex]
[tex]\sf : \; \implies 2 \bigg( cos^{4} \dfrac{\pi}{8} + cos^{4} \dfrac{3 \pi}{8} \bigg)[/tex]
[tex]\sf : \; \implies 2 \bigg( cos^{4} \dfrac{\pi}{8} + sin^{4} \bigg\{ \dfrac{ \pi}{2} - \dfrac{3 \pi}{8} \bigg\} \bigg)[/tex]
[tex]\sf : \; \implies 2 \bigg( cos^{4} \dfrac{\pi}{8} + sin^{4} \dfrac{ \pi}{8} \bigg)[/tex]
[tex]\sf : \; \implies \bigg\{ 2 \bigg( cos^{2} \dfrac{\pi}{8} + sin^{2} \dfrac{ \pi}{8} \bigg)^{2} - 2sin^{2} \dfrac{ \pi}{8} cos^{2} \dfrac{ \pi}{8} \bigg\}[/tex]
[tex]\sf : \; \implies 2 \bigg( 1 - \dfrac{ \bigg\{2sin \dfrac{\pi}{8} cos\dfrac{ \pi}{8} \bigg\}^{2}}{2} \bigg)[/tex]
[tex]\sf : \; \implies 2 \bigg( 1 - \dfrac{ sin \bigg\{2 \times \dfrac{\pi}{8} \bigg\}^{2}}{2} \bigg)[/tex]
[tex]\sf : \; \implies 2 - sin^{2}( \pi - 4 )[/tex]
[tex]\sf : \; \implies 2 - \bigg( \dfrac{1}{\sqrt{2}} \bigg)^{2}[/tex]
[tex]\sf : \; \implies 2 - \dfrac{1}{2}[/tex]
[tex]\sf : \; \implies \dfrac{(2 \times 2 )-1}{2}[/tex]
[tex]\sf : \; \implies \dfrac{4-1}{2}[/tex]
[tex]\red{\large{\boxed{\boxed{ \bf \leadsto \dfrac{3}{2} \; }}}}}[/tex]