Question

If cos alpha by cos beta = m and cos alpha by sin beta = n find ( m^2 + n^2)cos ^2 beta = n^2

Answer 1

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Answer 2

Answer:

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

Step-by-step explanation:

$Given \: \frac{cos\alpha}{cos\beta}=m$----(1)

$\frac{cos\alpha}{sin\beta}=n$----(2)

Now ,

$LHS = (m^{2}+n^{2})cos^{2}\beta}\\=[\big(\frac{cos\alpha}{cos\beta}\big)^{2}+\big(\frac{cos\alpha}{sin\beta}\big)^{2}]cos^{2}\beta\\=[\frac{cos^{2}\alpha}{cos^{2}\beta}+\frac{cos^{2}\alpha}{sin^{2}\beta}]cos^{2}\beta}\\=[cos^{2}\alpha\big(\frac{1}{cos^{2}\beta}+\frac{1}{sin^{2}\beta}\big)]cos^{2}\beta\\=[\frac{sin^{2}\beta+cos^{2}\beta}{cos^{2}\beta sin^{2}\beta}]\times cos^{2}\alpha\times cos^{2}\beta$

/* by Trigonometric identity:

sin²A+cos²A = 1

*/

=$\frac{cos^{2}\alpha\times cos^{2}\beta}{cos^{2}\beta sin^{2}\beta}$

=$\frac{cos^{2}\alpha}{sin^{2}\beta}$

=$\big(\frac{cos\alpha}{sin\beta}\big)^{2}$

= $n^{2}$

= $RHS$

Therefore,

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

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