Question

if x = cot theta + tan theta ; y= sec theta - cos theta eliminate theta from the equations

Answer 1

Given : x = cotθ + tanθ and y = secθ - cosθ

we have to eliminate θ from the equations.

solution : x = cotθ + tanθ

⇒x = cosθ/sinθ + sinθ/cosθ

= (cos²θ + sin²θ)/sinθ cosθ

= 1/(sinθ cosθ) ........(1)

y = secθ - cosθ = 1/cosθ - cosθ

= (1 - cos²θ)/cosθ

= sin²θ/cosθ ........ (2)

dividing equation (2) by (1) we get,

y/x = [sin²θ/cosθ]/[1/sinθ cosθ]

⇒y/x = sin³θ

⇒sinθ = $\sqrt[3]{\frac{y}{x}}$

from equation (1) we get,

x = 1/($\sqrt[3]{\frac{y}{x}}$cosθ)

⇒cosθ = 1/x$\sqrt[3]{\frac{y}{x}}$

⇒cosθ = 1/$\sqrt[3]{x^2y}$

now using identity, sin²θ + cos²θ = 1

⇒$\left(\sqrt[3]{\frac{y}{x}}\right)^2+\left(\frac{1}{\sqrt[3]{x^2y}}\right)^2=1$

⇒$\left(\frac{y}{x}\right)^{2/3}+\left(\frac{1}{x^2y}\right)^{2/3}=1$

Therefore the required equation is $\left(\frac{y}{x}\right)^{2/3}+\left(\frac{1}{x^2y}\right)^{2/3}=1$

Answer 2

Hope you find this answer helpful.