Question

If cos alpha by cos beta equals to M and cos alpha by sin beta equals to n then prove that (m square + n square) cos square beta equals to n square

Answer 1

correct or not I am not sure

Answer 2

Answer:

Step-by-step explanation:

Given

$\frac{cos\alpha}{cos\beta }=m$

$\frac{cos\alpha}{sin\beta }=n$

Hence prove that  

$(m^{2}+n^{2})cos^{2}\beta=n^{2}$

LHS

$(m^{2}+n^{2})cos^{2}\beta\\\\\Rightarrow\((\frac{cos\alpha}{cos\beta})^{2}+(\frac{cos\alpha}{sin\beta})^{2})cos^{2}\beta [\text{putting tha value of m and n we get}]$

$\\\\\Rightarrow(\frac{cos^{2}\alpha}{cos^{2}\beta}+\frac{cos^{2}\alpha}{sin^{2}\beta})cos^{2}\beta\\\\\Rightarrow(\frac{cos^{2}\alpha\times{sin}^{2}\beta+cos^{2}\alpha\times{cos}^{2}\beta}{cos^{2}\beta\times{sin}^{2}\beta})cos^{2}\beta\\\\\Rightarrow(\frac{cos^{2}\alpha(sin^{2}\beta+{cos}^{2}\beta)}{cos^{2}\beta\times{sin}^{2}\beta})cos^{2}\beta$

$\\\\\Rightarrow(\frac{cos^{2}\alpha}{cos^{2}\beta\times{sin}^{2}\beta})cos^{2}\beta [\text {we know that}\text{ } sin^{2}\beta+cos^{2}\beta=1}]\\\\\Rightarrow\frac{cos^{2}\alpha}{sin^{2}\beta}\\\\\Rightarrow{n}^{2}$

Hence LHS=RHS (Proved)