Question

cos^4(7Pi/8)+cos^4(3Pi/8)=

Answer 1

Cos^4(7Pi/8)+cos^4(3Pi/8)=Cos^4(8Pi/8 - 1Pi/8)+cos^4(4Pi/8 - 1Pi/8)=
=Cos^4(Pi - 1Pi/8)+cos^4(Pi/2 - 1Pi/8)=Cos^4(Pi/8)+sin^4(Pi/8) = 
 = ((1+cosPi/4)/2)^2+((1-cosPi/4)/2)^2 =[ (1+cosPi/4)^2+(1-cosPi/4)^2 ] /4= 
= [1+2cosPi/4+(cosPi/4)^2+2cosPi/4+(cosPi/4)^2] / 4 = 
 = [2+2(cosPi/4)^2 ] / 4 = [2+2( 1/√2)^2 ] / 4 = [2+1 ] / 4 = 3/4

Ответ: 3/4

Answer 2

$Cos ^{4} \frac{7 \pi }{8} +Cos ^{4} \frac{3 \pi }{8} =Cos ^{4}( \pi - \frac{ \pi }{8})+Cos ^{4} ( \frac{ \pi }{2}- \frac{ \pi }{8}) = Cos ^{4} \frac{ \pi }{8} +Sin ^{4} \frac{\pi }{8}=$$=(Cos ^{2} \frac{ \pi }{8} ) ^{2}+(Sin ^{2} \frac{ \pi }{8}) ^{2}=( \frac{1+Cos \frac{ \pi }{4} }{2} ) ^{2} +( \frac{1-Cos \frac{ \pi }{4} }{2} ) ^{2} =$$= \frac{1+2Cos \frac{ \pi }{4}+Cos ^{2} \frac{ \pi }{4}+1-2Cos \frac{ \pi }{4} +Cos ^{2} \frac{ \pi }{4} }{4}= \frac{2+2Cos ^{2} \frac{ \pi }{4} }{4} = \frac{1+( \frac{1}{ \sqrt{2} } ) ^{2} }{2} = \frac{1,5}{2}=0,75$