If x-asin- and y-btan- then prove that a2x2-b2y2-1-

x=asinθ ∴ax=θ ∴ y=btanθ ∴by=θ ∴a^2x^2 b^2y^2=^2θ ^2θ ∴a^2x^2 b^2y^2=1 ...

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Standard X

Mathematics

Trigonometric Identities

If x=asinθ ...

Question

If $x = a sin θ$ and $y = b tan θ$ , then prove that $\frac{a^{2}}{x^{2}} - \frac{b^{2}}{y^{2}} = 1$ .

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Solution

$x = a sin θ$
$∴ \frac{a}{x} = csc θ$
$∴ y = b tan θ$
$∴ \frac{b}{y} = cot θ$
$∴ \frac{a^{2}}{x^{2}} - \frac{b^{2}}{y^{2}} = {csc}^{2} θ - {cot}^{2} θ$
$∴ \frac{a^{2}}{x^{2}} - \frac{b^{2}}{y^{2}} = 1$

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ii)

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$If \frac{x}{a} c o s θ + \frac{y}{b} s i n θ = 1 and \frac{x}{a} s i n θ - \frac{y}{b} c o s θ = 1, prove that \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 2$

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If x=asinθ and y=btanθ, then prove that a2x2−b2y2=1.

x=asinθ∴ax=cscθ∴y=btanθ∴by=cotθ∴a2x2−b2y2=csc2θ−cot2θ∴a2x2−b2y2=1

If $x = a sin θ$ and $y = b tan θ$ , then prove that $\frac{a^{2}}{x^{2}} - \frac{b^{2}}{y^{2}} = 1$ .

$x = a sin θ$
$∴ \frac{a}{x} = csc θ$
$∴ y = b tan θ$
$∴ \frac{b}{y} = cot θ$
$∴ \frac{a^{2}}{x^{2}} - \frac{b^{2}}{y^{2}} = {csc}^{2} θ - {cot}^{2} θ$
$∴ \frac{a^{2}}{x^{2}} - \frac{b^{2}}{y^{2}} = 1$