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As I'm working on my trigonometry exercises, I come across an exercise called "prove the equality," which I proceed to solve. This all goes well, until I arrive at a certain point, after which I can't solve any further and don't understand the explanation offered in the answers section. If you would please help me understand the way of thinking needed to solve this question, I'd be very happy!

Here's the problem:

Solve the equality:

$$\frac1{\tan(x)} + \tan(x) = \frac1{\sin(x)\cdot\cos(x)}$$

And here's what I've solved for:

$$\text{RHS}=\frac{1}{\frac{\sin(x)}{\cos(x)}} + \frac{\sin(x)}{\cos(x)} =\frac{\cos x}{\sin x} +\frac{\sin(x)}{\cos(x)}$$

Then, I get stuck. The answers section says the next step should be:

$$\frac{\cos^2(x) + \sin^2(x)}{\sin(x) \cdot \cos(x)} = \text{RHS}$$

I understand how this proves the equality, but I don't understand how they just went from the previous step to this one: where do the exponents suddenly come from??

If you would be so kind as to explain this sudden raise of powers to me... My gratitude will be infinite!

Thanks in advance
Lila

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Way of thinking: You want to simplify the more complicated side, which is the LHS, to a less complicated expression. You have two fractions with different denominators. So you multiply the numerator and denominators of the two fractions to obtain a common denominator.$$\frac1{\tan x}+\tan x=\frac{\cos x}{\sin x}+\frac{\sin x}{\cos x}=\frac{\cos^2 x}{\cos x\sin x}+\frac{\sin^2 x}{\sin x\cos x}=\frac{\cos^2 +\sin^2x}{\sin x\cos x}$$

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  • $\begingroup$ Thanks a lot! It's totally clear now, I feel kind of stupid for not getting it right away... $\endgroup$ – Lila May 3 '18 at 10:29
  • $\begingroup$ No problem! ${}$ $\endgroup$ – John Doe May 3 '18 at 10:30
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We simply have

$$\frac1{\tan x}+\tan x=\frac{\cos x}{\sin x}+\frac{\sin x}{\cos x}=\frac{\cos^2 x+\sin^2 x}{\sin x\cos x}$$

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