Question

1/sin18 - 1//sin54 =2

Answer 1

Answer with explanation:

We have to prove that

`(1)/(sin18)-(1)/(sin54)=2`

sin54°=sin3×18°=3 sin18°-4sin³18°

LHS

`(1)/(sin18^(\circ))-(1)/(3sin18^(\circ)-4sin^318^(\circ))`

`=(1)/(sin18^(\circ))*(1-(1)/(3-4 sin^218^(\circ)))\\\\=(1)/(sin18^(\circ))*((3-4sin^218^(\circ)-1)/(3-4 sin^2 18^(\circ)))\\\\=(1)/(sin18^(\circ))*((2-4sin^218^(\circ))/(3-4 sin^2 18^(\circ)))\\\\=(2)/(sin18^(\circ))*((1-2sin^218^(\circ))/(3-4 sin^2 18^(\circ)))\\\\=(2cos36^(\circ))/(sin54^(\circ))`

`=2(sin54^(\circ))/(sin54^(\circ))\\\\=2`

Hence Proved.

Used the following trigonometric Identity to solve the problem

Sin3A=3 SinA-4 sin³A

Sin(90°-A)=Cos A &Cos(90°-A)=SinA

Cos 2A=2Cos²A-1=1-2Sin²A=Cos²A-Sin²A

Answer 2

We are given with the equation
1/sin 18 - 1/sin 54 = 2
We are meant to verify if the equation is true
So,
(sin 54 - sin 18)/ sin 18 sin 54 = 2
sin 54 - sin 18 = 2 sin 18 sin 54
sin 18*3 - sin 18 = 2 sin 18 sin 18*3
Use trigonometric identities to verify the equation.
The identities to use are the double and product of angles.