Question

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

Answer 1

Answer:  see proof below

Step-by-step explanation:

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

                                                                        sin 2A = 2sinA · cosA

$\text{Given:}\quad \tan A = \dfrac{x}{y}$

Proof LHS = RHS

y cos 2A + x sin 2A = y

                  x sin 2A = y - y cos 2A

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

                  $\dfrac{x}{y}=\dfrac{1-\cos2A}{sin\ 2A}$

                  $\dfrac{x}{y}=\dfrac{1-(1-2sin^2A)}{2\sin A\cdot \cos A}$

                  $\dfrac{x}{y}=\dfrac{2sin^2A}{2\sin A\cdot \cos A}$

                  $\dfrac{x}{y}=\dfrac{\sin A}{\cos A}$

                  $\dfrac{x}{y}=\tan A}$

This is a TRUE statement since it was given that  $\tan A = \dfrac{x}{y}$