Question

cos square 3 pi by 5 + cos square 4 pi by 5 equal to what

Answer 1

Answer:

Cos^2 3π/5 + Cos^2 4π/5

= Cos^2 108 + Cos^2 144

=Cos^2(90+18) + Cos^2(90+54)

= Sin^2 18 + Sin^2 54

= (√5+1/4)^2 + (√5-1/4)^2

[ a+b]^2 - [ a-b]^2 = 2(a^2 + b^2)

=(√5+1)^2 + √(5-1)^2/16

=2(5+1)/16

=6/8

= 3/4

Answer 2

Given :$Cos^2 \frac{3\pi}{5} + Cos^2 \frac{4 \pi}{5}$

To find : Solve

Solution:

$Cos^2 \frac{3\pi}{5} + Cos^2 \frac{4 \pi}{5}$

$Cos^2 108 + Cos^2 144$

$Cos^2(90+18) + Cos^2(90+54)$

Identity: $Cos (90+\theta)=-sin \theta$

$Sin^2 18 +Sin^2 54\\=(\frac{1}{4}(\sqrt{5}-1))^2+(\frac{1}{4}(\sqrt{5}+1))^2\\= (\sqrt{5}+\frac{1}{4})^2 + (\sqrt{5}-\frac{1}{4})^2$

Identity :$(a+b)^2=a^2+b^2+2ab$

$(a-b)^2=a^2+b^2-2ab$

Using identity :

$=5+\frac{1}{16}+\frac{2\sqrt{5}}{4}+5+\frac{1}{16}-\frac{2\sqrt{5}}{4}\\\\=\frac{81}{8}$

Hence$Cos^2 \frac{3\pi}{5} + Cos^2 \frac{4 \pi}{5}=\frac{81}{8}$