Simplify the trigonometric expression sin(x)^2+(sin(x)^4)/(cos(x)^2)

 Exercise

$sen^2\left(x\right)+\left(\frac{sen^4\left(x\right)}{cos^2\left(x\right)}\right)$

 

Combine all terms into a single fraction with $\cos\left(x\right)^2$ as common denominator

$\frac{\sin\left(x\right)^2\cos\left(x\right)^2+\sin\left(x\right)^4}{\cos\left(x\right)^2}$

 

Factor the polynomial $\sin\left(x\right)^2\cos\left(x\right)^2+\sin\left(x\right)^4$ by it's greatest common factor (GCF): $\sin\left(x\right)^2$

$\frac{\sin\left(x\right)^2\left(\cos\left(x\right)^2+\sin\left(x\right)^2\right)}{\cos\left(x\right)^2}$

 

Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$

$\frac{\sin\left(x\right)^2}{\cos\left(x\right)^2}$

 Why is sin(x)^2 + cos(x)^2 = 1 ?

 

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

$\tan\left(x\right)^2$

 Why does (sin(x)^n)/(cos(x)^n) = tan(x)^n ?

$\tan\left(x\right)^2$

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