Question

That sin 4 pi by 8 + sin 3 pi by 8 + sin 45 by 8 + 9 4 7 x 8 is equal to 3 by 2

Answer 1

Answer:

Step-by-step explanation:

Let us use the trigonometric identity  using this identity let us convert

Sin⁴π/8 + Sin⁴3π/8 + Sin⁴5π/8+Sin⁴7π/8

= Sin⁴π/8 + Sin⁴3π/8 + Sin⁴(π - 3/8)+Sin⁴ (π - π/8)

= Sin⁴π/8 + Sin⁴3π/8+Sin⁴3π/8+Sin⁴π/8 (∵Sin(π-θ) = Sinθ)

= 2(Sin⁴π/8 + Sin⁴3π/8)

= 2[Sin⁴π/8+Cos⁴(π/2 - 3π/8)]  (∵ Cos(π/2 - θ) =Sinθ)

=  2[Sin⁴π/8+ Cos⁴π/8]

= 2[ (Sin²π/8 + Cos²π/8)²  - 2Sin²π/8Cos²π/8]

= 2[ 1 - (Sin2(π/8))²/2]  (∵ Sin2θ =2SinθCosθ)

= 2[ 1 - 1/2(Sin²π/4)]

= 2[1 - 1/4] (∵ Sinπ/4 =1/√2)

= 2*3/4 = 3/2.

= R.H.S

Hence proved.