Question

Prove the following identity : If x = a sec θ, y = b tan θ, prove that a^2/x^2-b^2/y^2=1​

Answer 1

Answer:

$x = a \: sin \: θ \\ \frac{x}{a} = \sin \: θ \\ \frac{a}{x} = \frac{1}{sin \: θ} \\ squaring \: on \: both \: the \: sides \\ { (\frac{a}{x} )}^{2} = {( \frac{1}{sin \: θ} )}^{2} \\ \frac{ {a}^{2} }{ {x}^{2} } = \frac{1 }{ {sin}^{2}θ }$

$y = b \: tan \: θ \\ \frac{y}{b} = tan \: θ \\ \frac{b}{y} = \frac{1}{tan \: θ} \\ squaring \: on \: both \: the \: sides \\ { (\frac{b}{y} )}^{2} = { (\frac{1}{tan \: θ} )}^{2} \\ \frac{ {b}^{2} }{ {y}^{2} } = \frac{1 }{ {tan}^{2} \: θ}$

$\frac{ {a}^{2} }{ {x}^{2} } - \frac{ {b}^{2} }{ {y}^{2} } = 1$

$\frac{1}{ {sin}^{2} θ} - \frac{1}{ {tan}^{2}θ } = 1$

$\frac{1}{ {sin}^{2}θ } - \frac{ {cos}^{2} θ}{ {sin}^{2}θ } = 1$

$\frac{1 - {cos}^{2}θ }{ {sin}^{2} θ} = 1$

$\frac{ {sin}^{2} θ}{ {sin}^{2}θ } = 1$

1=1

L.H.S=R.H.S

hence proved