Trig. Problem, totally stuck!

[yasaminG]

HI i need help with this problem, please help me if you can!!

@=theta

(sin^2@)(1+ncot^2@)

we need to prove that it is equal to (cos^2@)(n+tan^2@)

 

[soroban]

Hello, yasaminG!

$\displaystyle \sin^2\theta(1\,+\,n\cdot\cot^2\theta)\;=\;cos^2\theta(n\,+\,tan^2\theta)$

The left side is: .$\displaystyle \sin^2\theta\left(\frac{\cos^2\theta}{\cos^2\theta}\,+\,n\cdot\frac{\cos^2\theta}{\sin^2\theta}\right)\;=\;\frac{\sin^2\theta\cdot\cos^2\theta}{\cos^2\theta}\,+\,n\cdot\cos^2\theta$

Factor: .$\displaystyle \cos^2\theta\left(\frac{sin^2\theta}{\cos^2\theta}\,+\,n\right)\;=\;\cos^2\theta(\tan^2\theta\,+\,n)$

 

[yasaminG]

thank you so much!!