Question

If $\tan \theta =\frac{a}{b}$, prove that $\frac{a\sin \theta +b\cos \theta }{a\sin \theta +b\cos \theta }=\frac{a^2-b^2}{a^2+b^2}$.

Answer 1

Given:

…… (1)

Now, we know that

Therefore equation (1) becomes

…… (2)

Now, multiplying by on both sides of equation (2)

We get,

Therefore,

…... (3)

Now by applying dividendo in above equation (3)

We get,

…… (4)

Now by applying componendo in equation (3)

We get,

…… (5)

Now, by dividing equation (4) by equation (5)

We get,

Therefore,

Therefore, and cancels on L.H.S and R.H.S respectively and we get,

Hence, it is proved that