Question

x=asin¢ and y=btan¢ prove asquare/xspuqre-bsquare/ysquare=1

Answer 1

If x=asin¢ and y=btan¢ then
$\frac{a}{x} = \frac{1}{ \sin(c) } \: \: \: \: and \: \: \frac{b}{y} = \frac{1}{ \tan(c) }$
Where ¢=c
So.
$\frac{a}{x} = \csc(c) \: \: \: \: and \: \: \: \: \frac{b}{y} = \cot(c) \\ so.... \\ \frac{ {a}^{2} }{ {x}^{2} } = { \csc(c) }^{2} \: \: \: \: and \: \frac{ {b}^{2} }{ {y}^{2} } = { \cot(c) }^{2} \\ thus... \\ \frac{ {a}^{2} }{ {x}^{2} } - \frac{ {b}^{2} }{ {y}^{2} } = { \csc(c) }^{2} - { \cot(c) }^{2} = 1 (proved)\\ since ... \: \: { \csc(c) }^{2} - { \cot(x) }^{2} = 1$