Question

if Alfa =22°30' then (1+cos alfa)(1+cos3alfa)(1+cis5alfa)(1+cos7alfa)

Answer 1

Hello there
Here alpha=22.5°=π/8
or,(1+cosπ/8)(1+cos3π/8)(1+cos5π/8)(1+cos7π/8)
or,(1+cosπ/8)(1+cos3π/8)(1+cos(π-3π/8))(1+cos(π-π/8))
or,(1+cosπ/8)(1-cosπ/8)(1+cos3π/8)(1-cos3π/8)
or,(1-cos²π/8)(1-cos²3π/8)
or,sin²π/8 x sin²3π/8
Multiplying and dividing the equation by 4 we get
(4sin²π/8 x sin²3π/8)/4
or,(2sinπ/8 xsin3π/8)²/4
or,(cos(3π/8-π/8)-cos(3π/8+π/8))²/4
or,(cosπ/4-cosπ/2)²/4
or,(1/√2+0)²/4
or,1/16. Answer
Hope it helps.

Answer 2

Answer:

1/8

Step-by-step explanation:

Given,

α = 22°30′ = 22 1/2° = 45°/2=π/4×2 = π/8

Now,

(1 + cos α)(1 + cos 3α)(1 + cos 5α)(1 + cos 7α)

= (1 + cos π/8) (1 + cos 3π/8)(1 + cos 5π/8)(1 + cos 7π/8)

= (1 + cos π/8)(1 + cos 3π/8)[1 + cos(π – 3π/8)] [1 + cos(π – π/8)]

= (1 + cos π/8)(1 + cos 3π/8)(1 – cos 3π/8)(1 – cos π/8)

= [1 – cos2(π/8)] [1 – cos2(3π/8)]

= sin2(π/8) sin2(3π/8)

= (sin π/8 sin 3π/8)2

Using the formula sin x sin y = (1/2)[cos(x – y) – cos(x + y)],

= [(1/2){cos(π/8 – 3π/8) – cos(π/8 + 3π/8)}]2

= [(1/2)(cos π/4 – cos π/2)]2

= [(1/2){(1/√2) – 0}]2

= (1/4) (1/2)

= 1/8