Prove the trigonometric identity sin(x)-sin(x)^3=sin(x)cos(x)^2

 Exercise

$\sin\left(x\right)-\sin\left(x\right)^3=\sin\left(x\right)\cos\left(x\right)^2$

 

Starting from the left-hand side (LHS) of the identity

$\sin\left(x\right)-\sin\left(x\right)^3$

 

Factor the polynomial $\sin\left(x\right)-\sin\left(x\right)^3$ by it's greatest common factor (GCF): $\sin\left(x\right)$

$\sin\left(x\right)\left(1-\sin\left(x\right)^2\right)$

 

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

$\sin\left(x\right)\cos\left(x\right)^2$

 Why is 1 - sin(x)^2 = cos(x)^2 ?

 

Since we have reached the expression of our goal, we have proven the identity

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