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1

Solved example of special quotients

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

I. Choose what side of the identity to work on

2

To prove an identity, we usually begin to work on the side of the equality that seems to be more complicated. In this case, we will choose to work on the left side $\frac{\cos\left(x\right)^2-\sin\left(x\right)^2}{\cos\left(x\right)+\sin\left(x\right)}$ to reach the right side $\cos\left(x\right)-\sin\left(x\right)$

II. Express in terms of sine and cosine

III. Operate, group, simplify

3

The difference of the squares of two terms, divided by the sum of the same terms, is equal to the difference of the terms, in other words:

  • $\displaystyle\frac{a^2-b^2}{a+b}=a-b$
  • Where the value of $a$ is $\cos\left(x\right)$
  • and the value of $b$ is $\sin\left(x\right)$, therefore:

$\cos\left(x\right)-\sin\left(x\right)=\cos\left(x\right)-\sin\left(x\right)$

IV. Check if we arrived at the expression we wanted to prove

4

Both expressions are equal

true

Answer

true

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