Question

If tan tetha = Q sin theta/P+Q cos alpha prove that tan (alpha- theta)= P sin alpha/Q+P cos alpha

Answer 1

Given that,

If $\tan\theta=\dfrac{Q\sin\alpha}{P+Q\cos\alpha}$

The given function is

$\tan(\alpha-\theta)=\dfrac{P\sin\alpha}{Q+P\cos\alpha}$

We need to prove the given function

Using left hand side

$=\tan(\alpha-\theta)$

$=\dfrac{\tan\alpha-\tan\theta}{1+\tan\alpha\tan\theta}$

$=\dfrac{\tan\alpha-\dfrac{Q\sin\alpha}{P+Q\cos\alpha}}{1+\tan\alpha\times\dfrac{Q\sin\alpha}{P+Q\cos\alpha}}$

$=\dfrac{P\tan\alpha+Q\tan\alpha\cos\alpha-Q\sin\alpha}{P+Q\cos\alpha+\tan\alpha\times Q\sin\alpha}$

$=\dfrac{P\tan\alpha+Q\sin\alpha-Q\sin\alpha}{P+Q\cos\alpha+\dfrac{Q\sin^2\alpha}{\cos\alpha}}$

$=\dfrac{P\sin\alpha}{\cos\alpha(P+Q\cos\alpha+\dfrac{Q\sin^2\alpha}{\cos\alpha})}$

$=\dfrac{P\sin\alpha}{P\cos\alpha+Q\cos^2\alpha+Q\sin^2\alpha}$

$=\dfrac{P\sin\alpha}{P\cos\alpha+Q}$

$=R.H.S$

Hence, This is proved