Question

Find the value of :-
$\bf \cos^4\dfrac{\pi}{8} +\cos^4\dfrac{3\pi}{8} + \cos^4\dfrac{5\pi}{8} + \cos^4 \dfrac{7\pi}{8}$

Answer 1

$\bf\huge{\pink{Question}}$

What is the value of cos^4 (pi/8) +cos^4 (3pi/8) +cos^4 (5pi/8) +cos^4 (7pi/8)?

$\bf\huge{\pink{Answer}}$

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

Answer 2

Answer :-

• The value is 3/2

Solution :-

$\sf : \; \implies cos^{4} \dfrac{\pi}{8} + cos^{4} \dfrac{3 \pi}{8} + cos^{4} \dfrac{5 \pi}{8} + cos^{4} \dfrac{ 7 \pi}{8}$

$\sf : \; \implies cos^{4} \dfrac{\pi}{8} + cos^{4} \dfrac{3 \pi}{8} + cos^{4} \bigg( \pi - \dfrac{3\pi}{8} \bigg) + cos^{4} \bigg( \pi - \dfrac{ \pi}{8} \bigg)$

$\sf : \; \implies cos^{4} \dfrac{\pi}{8} + cos^{4} \dfrac{3 \pi}{8} + cos^{4} \dfrac{3 \pi}{8} + cos^{4} \dfrac{ \pi}{8}$

$\sf : \; \implies 2 \bigg( cos^{4} \dfrac{\pi}{8} + cos^{4} \dfrac{3 \pi}{8} \bigg)$

$\sf : \; \implies 2 \bigg( cos^{4} \dfrac{\pi}{8} + sin^{4} \bigg\{ \dfrac{ \pi}{2} - \dfrac{3 \pi}{8} \bigg\} \bigg)$

$\sf : \; \implies 2 \bigg( cos^{4} \dfrac{\pi}{8} + sin^{4} \dfrac{ \pi}{8} \bigg)$

$\sf : \; \implies \bigg\{ 2 \bigg( cos^{2} \dfrac{\pi}{8} + sin^{2} \dfrac{ \pi}{8} \bigg)^{2} - 2sin^{2} \dfrac{ \pi}{8} cos^{2} \dfrac{ \pi}{8} \bigg\}$

$\sf : \; \implies 2 \bigg( 1 - \dfrac{ \bigg\{2sin \dfrac{\pi}{8} cos\dfrac{ \pi}{8} \bigg\}^{2}}{2} \bigg)$

$\sf : \; \implies 2 \bigg( 1 - \dfrac{ sin \bigg\{2 \times \dfrac{\pi}{8} \bigg\}^{2}}{2} \bigg)$

$\sf : \; \implies 2 - sin^{2}( \pi - 4 )$

$\sf : \; \implies 2 - \bigg( \dfrac{1}{\sqrt{2}} \bigg)^{2}$

$\sf : \; \implies 2 - \dfrac{1}{2}$

$\sf : \; \implies \dfrac{(2 \times 2 )-1}{2}$

$\sf : \; \implies \dfrac{4-1}{2}$

$\red{\large{\boxed{\boxed{ \bf \leadsto \dfrac{3}{2} \; }}}}}$