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Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$ $$\tan x+\sec x=2\cos x$$

$$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=2\cos x$$ $$\left(\dfrac{\sin x+1}{\cos x}\right)=2\cos x$$ $$\sin x+1=2\cos^2x$$ $$2\cos^2x-\sin x+1=0$$
Edit: $$2\cos^2x=\sin x+1$$ $$2(1-\sin^2x)=\sin x+1$$ $$2\sin^2x+\sin x-1=0$$ $\sin x=a$ $$2a^2+a-1=0$$ $$(a+1)(2a-1)=0$$ $$a=-1,\dfrac{1}{2}$$ $$\arcsin(-1)=-90^\circ=-\dfrac{\pi}{2}$$ $$\arcsin\left(\dfrac{1}{2}\right)=30^\circ=\dfrac{\pi}{6}$$ $$180^\circ-30^\circ=150^\circ=\dfrac{5 \pi}{6}$$ $$x=\dfrac{\pi}{6},-\dfrac{\pi}{2},\dfrac{5 \pi}{6}$$ I actually do not know if those are the only answers considering my range is infinite:$-\infty\lt x\lt\infty$

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    This is incorrect; you all of a sudden dropped the term $\frac{1}{\cos(x)}$ from one side of your equation. Remember, anything you do to one side of an equation you must do to the other.2012-07-17
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    I thought I added it properly: $$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=\dfrac{\sin x}{\cos x}$$ same denominator? $$\dfrac{1}{3}+\dfrac{1}{3}=\dfrac{2}{3}$$2012-07-17
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    ${\sin x\over\cos x}+{1\over\cos x}={\sin x+1\over \cos x}$.2012-07-17
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    Okay, so how do I tackle the new problem. There are still two different functions.2012-07-17
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    Write the right hand side of $\sin x+1=2\cos^2 x$ as $2(1-\sin^2 x)$.2012-07-17
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    So where from here? Do I distribute the $2$ into the $(1-\sin^2x)$?2012-07-17
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    Yes. Then put everything on one side. Then think of it as a quadratic equation in the variable $\sin x$.2012-07-17
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    Please, check over my work to make sure it is correct.2012-07-17

5 Answers 5

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This is more a comment than an answer, but it is an important comment, since it affects the list of solutions. We seem to have solutions that come from $\sin x=\frac{1}{2}$, and solutions that come from $\sin x=-1$.

No problem with the $\sin x=\frac{1}{2}$ stuff. For the record, this gives the solutions $x=\frac{\pi}{6}+2k\pi$ and $x=\frac{5\pi}{6}+2k\pi$, where $k$ ranges over the integers, positive, negative, and $0$.

However, there is an issue with $\sin x=-1$. For then $\cos x=0$, and neither $\tan x$ nor $\sec x$ is defined. When we multiply through by $\cos x$, we must remember that if $\cos x=0$ we are multiplying both sides by $0$, so we are not getting an equivalent equation. Thus the "solutions" $-\frac{\pi}{2}+2k\pi$ must be discarded.

Remark: If we look at the function $\frac{\sin x+1}{\cos x}$, it turns out that it has a removable singularity at $x=-\frac{\pi}{2}$, and can be made continuous there by defining it to be $0$. In that sense, $x=-\frac{\pi}{2}$ is a solution. However, I doubt it would be accepted as a solution in the context the OP is in.

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    So, ultimately is $-\frac{\pi}{2}$ a real answer for the OP2012-07-17
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    @AustinBroussard: If it is high school, or early university, I am pretty sure that it would not be accepted as an answer. After all, if you "plug in" trying to compute $\tan(-\pi/2)$, the calculator will say **error** and refuse to do anything more until you reset it. Certainly the *process* by which it was obtained is flawed. One could make the removable discontinuity argument, which I did not write out. You would have to explain that the functions on the left are not defined at $-\frac{\pi}{2}$, but $\dots$. The safe thing is to explain why $-\frac{\pi}{2}$ is no good ($\tan$ not defined.)2012-07-17
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Hint 1: For all $x$, $\sin^2x+\cos^2x=1$
Hint 2: For $at^2+bt+c=0$ the solutions are $t_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

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You correctly applied the definitions of $\tan$ and $\sec$ to go from the equation in the problem statement to $$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=2\cos x.$$ However, you combined the two fractions incorrectly; in general $$\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$$ For example, you know that $$\frac{1}{4}+\frac{1}{4}=\frac{2}{4}=\frac{1}{2}$$ and that $$\frac{1}{4}+\frac{1}{4}\neq\frac{1}{4}.$$


The correct form of the equation should be $$2\cos^2x-\sin x-1=0.$$ You will want to use the fact that for any $x$, $$\sin^2(x)+\cos^2(x)=1.$$

Thus, any occurrence of $\cos^2(x)$ can be replaced with $(1-\sin^2(x))$. $$2(1-\sin^2(x))-\sin(x)-1=0$$ $$[2\cdot1-2\cdot\sin^2(x)]-\sin(x)-1=0$$ $$-2\cdot\sin^2(x)+(-1)\sin(x)-1+2=0$$ $$(-2)\sin^2(x)+(-1)\sin(x)+(1)=0$$

Then, you will need to use the quadratic formula to solve for $\sin(x)$. It may help to write $y=\sin(x)$ temporarily, to prevent confusion. Keep in mind that there may be two different possible values of $\sin(x)$ that result.

$$(-2)\sin^2(x)+(-1)\sin(x)+(1)=0$$ $$(-2)y^2+(-1)y+(1)=0$$ $$y=\frac{-(-1)\pm\sqrt{(-1)^2-4(-2)(1)}}{2(-2)}=\frac{1\pm\sqrt{1+8}}{-4}=\frac{1\pm 3}{-4}=\begin{cases}\frac{1+3}{-4}=\frac{4}{-4}=\fbox{$-1$} &\text{ or }\\\\ & \\ \frac{1-3}{-4}=\frac{-2}{-4}=\fbox{$\frac{1}{2}$}.\end{cases}$$

Lastly, you need to find those values of $x$ that produce that value (or those values) of $\sin(x)$. Keep in mind that the function $\sin(x)$ is periodic; here is part of its graph to illustrate: enter image description here
(and it continues similarly out to infinity in each direction) so that there are always infinitely many values of $x$ that will produce a given value of $\sin(x)$.

For any real number $x$, the values of $$\sin(x+2\pi k)$$ are all identical for any integer $k$,. There are usually even more values you can plug into $\sin$ that will give the same number (look at the graph for an idea).

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    I have fixed my mistake. Thank you for pointing that out. 48 hours of no sleep is getting to me. So excuse my mistakes.2012-07-17
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    @AustinBroussard For your own health, go and sleep.2012-07-17
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    I will after I finish a few more problems!2012-07-17
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    We have different answers! Darn..2012-07-17
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    I too hastily approved of your equation $$2\cos^2(x)-\sin(x)+1=0$$ In fact, the correct equation is $$2\cos^2(x)-\sin(x)-1=0$$ When you had $$2\cos^2(x)=\sin(x)+1$$ you neglected to distribute the negative sign: $$2\cos^2(x)-[\sin(x)+1]=0$$ $$2\cos^2(x)-\sin(x)-1=0$$.2012-07-17
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    Don't worry. That was your first mistake compared to all of mine. You're doing fine. And are my answers correct? Also read the very bottom about my limitations.2012-07-17
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    @Austin: The solution to the equation $2a^2+a-1=0$ corresponding to the factor $(a+1)$ is $a=-1$, not $a=1$.2012-07-17
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    I fixed it. I had the negative when I did: $\arcsin(-1)=-90^ \circ=-\dfrac{\pi}{2}$2012-07-17
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Of course, $x$ cannot be an integer multiple of $\pi$ plus $\frac{\pi}{2}$. We have here a quadratic equation after a little manipulation, which you, yourself, started.

$$\frac{\sin x}{\cos x}+\frac{1}{\cos x}=2\cos x$$ After multiplying by $\cos x$, using the identity $1=\sin^2 x+\cos^2 x$, and bringing everything to the left hand side: $$2\sin^2 x+\sin x-1=0$$ If we replace $y=\sin x$, then you will be able find a familiar expression for $y$: $$2y^2+y-1 = 0$$ Which gives, $y = \frac{1}{2}$ or $-1$. So, $\sin x$ is equal to either of those two values, which yields $x= 2n\pi+\frac{\pi}{6}$, $x=2n\pi+\frac{5\pi}{6}$ or $x=2n\pi+\frac{\pi}{2}$. However, we ruled out the latter solution initially.

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\begin{eqnarray} \left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)&=&2\cos x\\ \sin x + 1 &=& 2 \cos^2 x \\ \sin x + 1 &=& 2(1-\sin^2 x)\\ \end{eqnarray} Then $\sin x = -1$ or $\sin x = 1/2$ and the solutions are $-\pi + 2k \pi, \pi/6 + 2 k\pi$ and $5\pi/6 + 2k \pi$.

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    Can you explain the $+2k \pi$ part please.2012-07-17
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    Only for solutions out the $[-\pi,\pi]$2012-07-17