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cot θ - (csc^2 θ/cot θ) = -tan θ

 Mar 17, 2021
 #1
avatar+2111 
0

Being completly honest, I am not good with these things, but I'll give it my best shot. 

 

Let's start by turning everything into sine/cosine. 

(cos θ/sin θ) - (1/sin θ^2)/(cos θ/sin θ) = -(sine θ/cos θ)

(cos θ/sin θ) - (1/(sin θ * cos θ)) = -(sine θ/cos θ)

(cos θ^2 - 1)/(sin θ * cos θ) = -(sine θ/cos θ)

(cos θ^2 - 1) = (sin θ * cos θ) * -(sine θ/cos θ)

(cos θ^2 - 1) = - (sin θ^2)

 

cos θ = sqrt(1-sine θ^2)

This equation can be proven by using the unit circle and pythagreon theorum. 

Now, let's plug that into the original equation. 

 

(sqrt(1-sine θ^2)^2 - 1) = - (sin θ^2)

(1-sine θ^2 - 1) = - (sin θ^2)

- sine θ^2 = - (sin θ^2)

- sine θ^2 = - sine θ^2

 

omgosh, I can't believe I solved that. (hopefully it's correct)

I hope this helps. :)))

 

=^._.^=

 Mar 17, 2021
 #2
avatar+31 
+1

I hate to be this person but this problem has to be solved whilst keeping -tan alone on the right side

owentout  Mar 17, 2021
 #3
avatar+2111 
0

oh noessss. :(((((

 

Welp, at least I tried. 

Luckily, it looks like CPhill is typing so hopefully we can learn from him. :))

 

=^._.^=

catmg  Mar 17, 2021
 #4
avatar+31 
+2

I would very much hope so! Thank you for your gracious attempt though!

owentout  Mar 17, 2021
 #6
avatar+2111 
0

You're welcome :)))

Thank you for correcting me and responding. 

 

=^._.^=

catmg  Mar 17, 2021
 #5
avatar+121000 
+2

cot x  - ( csc^2 x  /cot x)                get a common denominator

 

cot ^2x   -csc^2  x

______________                 ( 1 + cot^2 x  = csc^2x   ⇒   cot^2 x   - csc^2 x  =  -1  )

    cot x

 

 

-1 /  cot x  =   

 

-1  / ( 1 /tan x) 

 

-1 (tan x)   =

 

-tan x

 

 

cool cool cool

 Mar 17, 2021
 #7
avatar+31 
+1

why does the cot^2 x happen after finding the common denominator?

owentout  Mar 17, 2021
 #8
avatar+121000 
+3

cot x            cot x                 cot^2 x

____  *     _______  =       _______

  1               cot x                cot x

 

So

 

cot ^2x         csc^2 x

_____  -     _______  =  

cot x            cot x

 

(cot^2 x  -csc^2 x)

_________________

         cot x

 

 

cool cool cool

CPhill  Mar 17, 2021
 #9
avatar+2111 
0

cot x - (csc^2 x / cot x)

cot^2 x / cot x = cot x

(cot^2 x / cot x) - (csc^2 x / cot x)

(cot^2 x - csc^2 x) / cot x

 

I learned that 1 + cot^2 x = csc^2 x today, thank you cphill. :)))

 

=^._.^=

catmg  Mar 17, 2021
 #10
avatar+121000 
+1

OK, catmg   !!!

 

 

cool cool cool

CPhill  Mar 17, 2021
 #11
avatar+31 
+2

This was an unnecessarily difficult problem, but thank you very much for the continuous assistance :) 

owentout  Mar 17, 2021
 #12
avatar+121000 
+2

They're  all  unnecessarily  difficult....until find out that many aren't    (LOL!!!)

 

 

cool cool cool

CPhill  Mar 17, 2021

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