Question

If $\alpha \csc A=p$ and $b\cot A=q$, then prove $\frac{p^2}{a^2}-\frac{q^2}{b^2}=1$

Answer 1

prove

$acosecA=p$ __ (i) prove $\frac{p^2}{a^2}-\frac{q^2}{b^2}=1$

$bcotA=q$ __ (ii)

$cosecA=\frac{p}{a}$ __ (iii)

$cotA=\frac{q}{b}$ __ (iv)

$\to$ square and substracted from equation (iii) & (iv)

$\frac{p^2}{a^2}-\frac{q^2}{b^2}=cose{c}^{2}A-co{t}^{2}A$

$=\frac{1}{si{n}^{2}A}-\frac{co{s}^{2}A}{si{n}^{2}A}$

$=\frac{1-co{s}^{2}A}{si{n}^{2}A}$

$=\frac{si{n}^{2}A}{si{n}^{2}A}$

$\therefore \frac{p^2}{a^2}-\frac{q^2}{b^2}=I$