Question

(a) If tan A=x/y
prove that: y.cos2A +x.sin2A= y.
Please help me..​

Answer 1

Answer: see proof below

Explanation:

Use the following Double Angle Identities: cos 2A = 1 - 2sin²A

sin 2A = 2sinA · cosA

`\text{Given:}\quad \tan A = (x)/(y)`

Proof LHS = RHS

y cos 2A + x sin 2A = y

x sin 2A = y - y cos 2A

x sin 2A = y(1 - cos 2A)

`(x)/(y)=(1-\cos2A)/(sin\ 2A)`

`(x)/(y)=(1-(1-2sin^2A))/(2\sin A\cdot \cos A)`

`(x)/(y)=(2sin^2A)/(2\sin A\cdot \cos A)`

`(x)/(y)=(\sin A)/(\cos A)`

`(x)/(y)=\tan A}`

This is a TRUE statement since it was given that
`\tan A = (x)/(y)`