Simplify the trigonometric expression (cos(x)^2-sin(x)^2)/(1+2sin(x)cos(x))

 Exercise

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

 

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

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

 Why is cos(2x) = cos(x)^2 - sin(x)^2 ?

 

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

$\frac{\cos\left(2x\right)}{1+2\cdot \left(\frac{1}{2}\right)\sin\left(2x\right)}$

 

Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)\sin\left(2x\right)$

$\frac{\cos\left(2x\right)}{1+\frac{2\cdot 1}{2}\sin\left(2x\right)}$

 

Multiply $2$ times $1$

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

 

Divide $2$ by $2$

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

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

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