1. ## Trig identities

Hi

Is there any way of converting the function below into one that doesn't involve a quotient?

$\displaystyle \frac{2\sec x}{2+\tan x}$

Thanks

2. Originally Posted by bobred
Hi

Is there any way of converting the function below into one that doesn't involve a quotient?

$\displaystyle \frac{2\sec x}{2+\tan x}$

Thanks
The expression simplifies to $\displaystyle \frac{2}{2 \cos x + \sin x}$ so I don't see a way of avoiding a quotient ....

3. Hello, bobred!

Is there any way of converting this function
. . into one that doesn't involve a quotient? . . $\displaystyle \frac{2\sec x}{2+\tan x}$
Yes, there is.
But the method may not be within the scope of your course.

We have: .$\displaystyle \frac{\dfrac{2}{\cos x}}{2 + \dfrac{\sin x}{\cos x}}$

Multiply by $\displaystyle \frac{\cos x}{\cos x}\!:\quad\frac{\cos x\left(\dfrac{2}{\cos x}\right)} {\cos x\left(2 + \dfrac{\sin x}{\cos x}\right)} \;=\;\frac{2}{2\cos x + \sin x}$ .[1]

Consider the denominator: .$\displaystyle 2\cos x + \sin x$

. . Multiply by $\displaystyle \tfrac{\sqrt{5}}{\sqrt{5}}\!:\quad \tfrac{\sqrt{5}}{\sqrt{5}}(2\cos x + \sin x) \;=\;\sqrt{5}\left(\tfrac{2}{\sqrt{5}}\cos x + \tfrac{1}{\sqrt{5}}\sin x\right)$

. . Let $\displaystyle \theta$ be an angle sucn that: .$\displaystyle \sin\theta = \tfrac{2}{\sqrt{5}},\;\cos\theta = \tfrac{1}{\sqrt{5}}$

. . Then we have: .$\displaystyle \sqrt{5}\left(\sin\theta\cos x + \cos\theta\sin x\right) \;=\;\sqrt{5}\sin(x + \theta)$

Hence, [1] becomes: .$\displaystyle \frac{2}{\sqrt{5}\sin(x + \theta)} \;=\;\frac{2}{\sqrt{5}}\csc(x + \theta)$

Therefore: .$\displaystyle \frac{2\sec x}{2 + \tan x} \;=\;\frac{2}{\sqrt{5}}\csc\bigg[x + \arccos\left(\tfrac{1}{\sqrt{5}}\right)\bigg]$

4. Now I see.

I had have been given some answers similar but none that have shown how they have arrived at the answer, Thanks

5. Just one more thing, why multipy by $\displaystyle \sqrt{5}$ and not something else?

6. Originally Posted by bobred
Just one more thing, why multipy by $\displaystyle \sqrt{5}$ and not something else?
Recall what Soroban wrote:
Let $\displaystyle \theta$ be an angle such that: .$\displaystyle \sin\theta = \tfrac{2}{\sqrt{5}},\;\cos\theta = \tfrac{1}{\sqrt{5}}$
You have a right triangle with angle $\displaystyle \theta$ where the opposite side is 2 and the adjacent side is 1. By the Pythagorean Theorem, the hypotenuse has to be $\displaystyle \sqrt{2^1 + 1^2} = \sqrt{5}$.

01

7. Thanks.

Thought it would be something obvious.