Question

If 5x = sec θ and $$\frac{5}{x}=\tan \theta$$find the value of $$5(x^2-\frac{1}{x^2})$$ 

Answer 1

Given: 

${\displaystyle 5x=\sec \theta ,\frac{5}{x}=\tan \theta }$ 

⇒ ${\displaystyle \sec \theta =5x,\tan \theta =\frac{5}{x}}$

We know that, 

${\displaystyle {\sec }^2\theta -{\tan }^2=1}$ 

⇒${\displaystyle {\left(5x\right)}^2-{\left(\frac{5}{x}\right)}^2=1}$ 

⇒ ${\displaystyle 25x^2-\frac{25}{x^2}=1}$  

⇒ ${\displaystyle 25(x^2-\frac{1}{x^2})=1}$

⇒${\displaystyle 5\times 5\times (x^2-\frac{1}{x^2})=1}$ 

⇒${\displaystyle 5(x^2-\frac{1}{x^2})=\frac{1}{5}}$