Can anyone simplify any of these questions?

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In summary, the conversation discusses simplifying questions and solving a quadratic equation. The specific problem being solved involves finding the value of $\tan\theta$ in the given equation.
  • #1
hmhmherath
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Can anyone simplify any of these questions?
 

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  • #2
Hello hmhmherath and welcome to MHB! :D

As per forum rules, in future please post at most two problems per thread.

Let's take these one at a time. For the first one (problem 11), do you know the formula for the sum of a geometric series?
 
  • #3
hmhmherath said:
Can anyone simplify any of these questions?

$\displaystyle \begin{align*} \frac{1}{\cos^2{(x)}} &= 2\,\sqrt{3}\tan{(x)}- 2 \\ 1 &= 2\,\sqrt{3} \sin{(x)}\cos{(x)} - 2\cos^2{(x)} \\ 1 &= \sqrt{3}\sin{(2\,x)} - 1 - \cos{(2\,x)} \\ 2 + 2\cos{(2\,x)} &= \sqrt{3}\sin{(2\,x)} \\ 4 + 8\cos{(2\,x)} + 4\cos^2{(2\,x)} &= 3 \sin^2{(2\,x)} \\ 4 + 8\cos{(2\,x)} + 4\cos^2{(2\,x)} &= 3 - 3\cos^2{(2\,x)} \\ 7\cos^2{(2\,x)} + 8\cos{(2\,x)} + 1 &= 0 \end{align*}$

Now solve the resulting quadratic.
 
  • #4
$$\frac{1}{\cos^2\theta}=2\sqrt3\tan\theta-2$$

$$\sec^2\theta=2\sqrt3\tan\theta-2$$

$$\tan^2\theta+1=2\sqrt3\tan\theta-2$$

$$\tan^2\theta-2\sqrt3\tan\theta+3=0$$

Can you solve that quadratic for $\tan\theta$ ?
 

FAQ: Can anyone simplify any of these questions?

Can you explain the concept in simpler terms?

Yes, I can try to simplify the concept by breaking it down into smaller and more easily understandable parts.

Is there a more straightforward way to understand this idea?

Yes, there may be other analogies or examples that can simplify the concept for better understanding.

Can you provide a real-life example to make it easier to grasp?

Yes, I can give an example or case study to illustrate the concept in action.

How can I apply this concept in everyday life?

Depending on the concept, I can suggest ways to apply it in practical situations or provide tips for implementing it in daily life.

Is there a visual aid that can simplify this concept?

Yes, I can create a visual representation such as a diagram or chart to help simplify the concept and make it easier to understand.

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