Simplify the trigonometric expression (sec(u)^2-1)/(sec(u)^2)

 Exercise

$\frac{\sec^2\left(u\right)-1}{\sec^2\left(u\right)}$

 

Expand the fraction $\frac{\sec\left(u\right)^2-1}{\sec\left(u\right)^2}$ into $2$ simpler fractions with common denominator $\sec\left(u\right)^2$

$\frac{\sec\left(u\right)^2}{\sec\left(u\right)^2}+\frac{-1}{\sec\left(u\right)^2}$

 

Simplify the resulting fractions

$1+\frac{-1}{\sec\left(u\right)^2}$

 

Applying the trigonometric identity: $\displaystyle\frac{1}{\sec^{n}(\theta)}=\cos^{n}(\theta)$

$1-\cos\left(u\right)^2$

 

Apply the trigonometric identity: $1-\cos\left(\theta \right)^2$$=\sin\left(\theta \right)^2$, where $x=u$

$\sin\left(u\right)^2$

 Why is 1 - cos(x)^2 = sin(x)^2 ?

$\sin\left(u\right)^2$

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