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TRIGONOMETRY HELP! Show that: tan30+1/tan30=1/sin30cos30

Mathematics
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\[\Large \tan x+\frac{1}{\tan x}=\frac{1}{\sin x \cosx} \] or for the LHS:\[\Large \frac{\tan^2x+1}{\tan x} \] Can you simplify this?
Yes? I think you can cancel out one tan and the x.
careful the x is a variable as in a function, you can never ever cancel that out, try to play with the numerator: \[\Large \tan^2(x)+1=? \] This is an identity you want to derive, and try to memorize, because it is all over the place.

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Other answers:

Why is Tan squared?
\[\Large \tan x+ \frac{1}{\tan x}=\frac{\tan^2(x)+1}{\tan(x)}\]
getting the same denominator. From Algebra.
Oh okay I see. So I have to simplify now. And I cancel out one tan therefore making it tan(x)+1/1?
So far I've input my value of 30 degrees to take the place of the x variable, Im now going to use the appropriate special triangle to find the value of tan30. Am I on the right track so far?
if you cancel out the tan you're left with the LHS of the equation, that's why they are equal.
As a general idea, that might improve your understanding for further such problems. Whenever you're dealt with problems that ask you to "verify" an identity, you want to manipulate the LHS or the RHS (usually you can choose, or it will jump into your eyes which one is easier) until you're capable to verify that they are both equal. LHS=RHS
So, since these are trig problems, you want to use trigonometric identities. For instance, it says: \[\Large \frac{\tan^2(x)+1}{\tan(x)} \] Of course you can factor and cancel things out here a bit, but that wont help you. it will just give you the original equation again, because that is where we started. Much rather, look at the numerator first, because it looks a little bit special doesn't it? \[\Large \tan^2(x)+1=? \] So, at first glimpse we don't know what that is. But since we see that we're in the world of trigonometry, we know that we want to use trigonometric identities and definitions. We know that: \[\Large \tan(x)=\frac{\sin(x)}{\cos(x)} \] So it's a good idea to substitute that back into the equation above.
Thank you so much you've been a great help. WIll it be possible for you to help me with another Trig related question?
Did you manage to verify this problem for yourself already?
Yes.
oh very good then!
Feel free to post further questions on OS anytime
This is a more application gears question. I struggle here the most. A baseball diamond forms a square of side length :27.4m. Sarah says she used a special triangle to calculate the distance between home plate and second base. Describe how Sarah might calculate this distance.
At this point I assumed that home plate is diagonally across from 2nd base. Therefore creating 2 triangles.

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