Simplify the trigonometric expression tan(x)^2-sin(x)^2

 Exercise

$tan^2x-sin^2x\:$

 

Apply the trigonometric identity: $\tan\left(\theta \right)^n$$=\frac{\sin\left(\theta \right)^n}{\cos\left(\theta \right)^n}$, where $n=2$

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

 Why is tan(x) = sin(x)/cos(x) ?

 

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

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

 

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

$\frac{\sin\left(x\right)^2\left(1-\cos\left(x\right)^2\right)}{\cos\left(x\right)^2}$

 

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

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

 Why is 1 - cos(x)^2 = sin(x)^2 ?

 

When multiplying exponents with same base we can add the exponents

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

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

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