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

 Exercise

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

 

Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$

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

 Why does sin(2x) = 2sin(x)cos(x) ?

 

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

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

 Why is 1 - sin(x)^2 = cos(x)^2 ?

 

Simplify $2\sin\left(x\right)\cos\left(x\right)$ using the trigonometric identity: $\sin(2x)=2\sin(x)\cos(x)$

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

 

Simplify the fraction $\frac{2\sin\left(2x\right)}{2}$ by $2$

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

 

Multiply the single term $2$ by each term of the polynomial $\left(1-\sin\left(x\right)^2\right)$

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

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

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