Simplify cot x tan x?

Question

So I'm completely blanking on an easy question like this, I can't see how such a small expression can be simplified any further. Can anyone help me out?

How would you go about simplifying something like (sin^2 u + tan^2 u + cos^2 u) / sec u?

I'm supposed to be using fundamental trigonometric identities in order to simplify them but I'm having a hard time applying them to the questions. Thanks for any help!

Answer 1

tan x=sin x/cos x

cot x=cos x/sin x so

tan x cot x=(sin x/cos x)*(cos x/sin x)=(sin x cos x)/(sin x cos x)=1



faster is cot x=1/tan x so

tan x cot x=tan x/tan x=1 but I thought you might like a little more detail.





(sin^2 u+tan^2 u+cos^2 u)/sec^2 u

sin^2 +cos^2=1 so

(1+tan^2 u)/sec^2 u sec=1/cos so

cos^2 u+cos^2 u sin^2 u/cos^2 u

cos^2 u+sin^2 u

1

Answer 2

You need to take a break. This one is soooo trivial.

cot(x)=1/tan(x)

cot(x)tan(x)=1



The other one is pretty straight forward too.

[sin²(u) + tan²(u) + cos²(u) ]/sec(u)

sin²(u) + cos²(u) = 1 so the numerator becomes

1+tan²(u) = 1+ sin²(u)/cos²(u) = [cos²(u)+sin²(u)]/cos²(u) = 1/cos²(u) = sec²(u)

So the rational expression is now

sec²(u)/sec(u)

=sec(u)



If you were to ask me how I came by that approach, I don't honestly know. I've done a few of these. I always try to convert everything into expressions involving sine and cosine whenever I can. I know I can express the rest of the functions in terms of these two function. That way I don't have to memorize a ton of identities.

Answer 3

note that cot x = 1/tan x...therefore the answer is 1.



for the next one, use Pythagorean Identities:



sin^2 x + cos^2 x = 1, and 1 + tan^2 x = sec^2 x, the rest is algebra.



I'll leave this all to you. good luck.

Answer 4

sin^2 x + cos^2 x = 1

sec^2 x = 1 + tan^2 x



(sin^2 u + tan^2 u + cos^2 u) / sec u

=(sin^2 u + cos^2 u + tan^2 u) / sec u

=(1 + tan^2 u)/sec u

= sec^2 u / sec u

= sec u







cot x = 1/(tan x)

(cot x) (tan x) = 1

Answer 5

sin squared plus cos squared is 1 plus cot squared(i think) so you would have 1/tansquare(cot square) + tang^2=1/sec^2= 1/1*cos/1=cos!!! i think.....???