Question

Please solve this problem ​

Answer 1

I think we have to show

$\frac{p \sin( \alpha ) - q \cos( \alpha ) }{p \sin( \alpha ) + q \cos( \alpha ) } = \frac{ {p}^{2} - {q}^{2} }{ {p}^{2} + {q}^{2} }$

This is correct .

we have given that ,

tan@=p/q

In LHS divide both numerator and denominator with cos@ we get ,

$\frac{p \tan( \alpha ) - q }{p \tan( \alpha ) + q}$

Now put the value of tan@=p/q

we get the solution

soln refers to the attachment

Answer 2

In LHS divide both Numerator and denominator by cos @

we get

ptan@-q/ptan@+q

now put the value of tan@

=p(p/q)-q/p(p/q)+q

=p^2/q-q/p^2/q+q

we get

=(p^2-q^2)/p^2+q^2