Question

If tan theta=m/n, prove that m sin theta- n cos theta / m sin theta + n cos theta=m× m - n×n /m ×m + n× n

Answer 1

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Answer 2

Answer and Explanation:

Given : $\tan\theta=\frac{m}{n}$

To prove : $\frac{m\sin\theta-n\cos\theta}{m\sin\theta+n\cos\theta}=\frac{m\times m-n\times n}{m\times m+n\times n}$

Solution :

We know,

$\tan\theta=\frac{P}{B}=\frac{m}{n}$

The hypotenuse is $H=\sqrt{P^2+B^2}$

$H=\sqrt{m^2+n^2}$

$\sin\theta=\frac{P}{H}=\frac{m}{\sqrt{m^2+n^2}}$

$\cos\theta=\frac{B}{H}=\frac{n}{\sqrt{m^2+n^2}}$

Substitute the values in LHS,

$\frac{m\sin\theta-n\cos\theta}{m\sin\theta+n\cos\theta}$

$=\frac{m(\frac{m}{\sqrt{m^2+n^2}})-n(\frac{n}{\sqrt{m^2+n^2}})}{m(\frac{m}{\sqrt{m^2+n^2}})+n(\frac{n}{\sqrt{m^2+n^2}})}$

$=\frac{\frac{m^2-n^2}{\sqrt{m^2+n^2}}}{\frac{m^2+n^2}{\sqrt{m^2+n^2}}}$

$=\frac{m^2-n^2}{m^2+n^2}$

$=\frac{m\times m-n\times n}{m\times m+n\times n}$

=RHS

LHS=RHS, Hence proved.