Question

If sinA=√3-1/2√2,then prove that cos2A=√3/2 prove that

Answer 1

`\boxed{\sf cos2A =(\sqrt3)/(2)}`

Explanation:

Here we are given that the value of sinA is √3-1/2√2 , and we need to prove that the value of cos2A is √3/2 .

Given :-

•
`\sf\implies sinA =(\sqrt3-1)/(2\sqrt2)`

To Prove :-

•
`\sf\implies cos2A =(\sqrt3)/(2)`

Proof :-

We know that ,

`\sf\implies cos2A = 1 - 2sin^2A`

Therefore , here substituting the value of sinA , we have ,

`\sf\implies cos2A = 1 - 2\bigg( (\sqrt3-1)/(2\sqrt2)\bigg)^2`

Simplify the whole square ,

`\sf\implies cos2A = 1 -2* ( 3 +1-2\sqrt3)/(8)`

Add the numbers in numerator ,

`\sf\implies cos2A = 1-2* (4-2\sqrt3)/(8)`

Multiply it by 2 ,

`\sf\implies cos2A = 1 - ( 4-2\sqrt3)/(4)`

Take out 2 common from the numerator ,

`\sf\implies cos2A = 1-(2(2-\sqrt3))/(4)`

Simplify ,

`\sf\implies cos2A = 1 -( 2-\sqrt3)/(2)`

Subtract the numbers ,

`\sf\implies cos2A = ( 2-2+\sqrt3)/(2)`

Simplify,

`\sf\implies \boxed{\pink{\sf cos2A =(\sqrt3)/(2)} }`

Hence Proved !