Question

Can anyone give step by step explanation for this ??pls make it quick...if the ans is relevant I’ll mark u as brainlist ...

Answer 1

Step-by-step explanation:

i). Sin(18⁰)=Sin(π/10)

Let x=π/10

⇒5x=π/2

⇒2x=π/2 -3x

⇒Sin(2x)=Sin(π/2 -3x)=Cos(3x)

⇒2Sinx.Cosx=4Cos³x-3Cosx

⇒2Sinx=4Cos²x-3

⇒2Sinx=4(1-Sin²x)-3

⇒2Sinx=4-4Sin²x-3

⇒4Sin²x+2Sinx-1=0

⇒Sinx={-2±√(4+16)}/8={-2±√(20)}/8={-2±2√5}/8=(-1±√5)/4

Neglecting -ve sign, we have,

Sinx=(√5 -1)/4

⇒Sin(π/10)=Sin(18⁰)=(√5 -1)/4

And Sin(54⁰)=Sin(3π/10)=3Sin(π/10)-4Sin³(π/10)=3.(√5 -1)/4-4{(√5 -1)/4}³=(3√5 -3)/4 -(5√5 -1+3√5-15)/16=(12√5-12-8√5+16)/16=(4√5+4)/16=(√5+1)/4

₀⁰₀ Sin(18⁰)Sin(54⁰)={(√5 -1)/4}*{(√5 +1)/4}=(5-1)/16=4/16=1/4

ii). 2Cosec(9⁰)Sec(9⁰)

Cos(π/10)=Cos(18⁰)=√(1-Sin²(π/10))=√[1-{(√5-1)/4}²]=√{1-(5+1-2√5)/16}=√{(16-6+2√5)/16}={√(10+2√5)}/4

Sin(9⁰)=Sin(π/20)=√[{1-Cos(π/10)}/2]=√[{1-{√(10+2√5)}/4}/2]=√[{4-√(10+2√5)}/8]

Cosec(9⁰)=√[8/{4-√(10+2√5)}]

Cos(9⁰)=√{1-Sin²(9⁰)}=√[1-{4-√(10+2√5)}/8]=√[{4+√(10+2√5)}/8]

Sec(9⁰)=√[8/{4+√(10+2√5)}]

₀⁰₀ 2Cosec(9⁰)Sec(9⁰)=2*√[8/{4-√(10+2√5)}] *√[8/{4+√(10+2√5)}]=2*8/√{16-10-2√5}=16/√(6-2√5)

identity- Cos(2x)=1-2Sin²x

⇒Sin²x={1-Cos(2x)}/2

⇒Sinx=√[{1-Cos(2x)}/2]