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# Prove/ Verify the given identity

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cot θ - (csc^2 θ/cot θ) = -tan θ

Mar 17, 2021

#1
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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
+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
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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
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I would very much hope so! Thank you for your gracious attempt though!

owentout  Mar 17, 2021
#6
+2111
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You're welcome :)))

Thank you for correcting me and responding.

=^._.^=

catmg  Mar 17, 2021
#5
+121000
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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

Mar 17, 2021
#7
+31
+1

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

owentout  Mar 17, 2021
#8
+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

CPhill  Mar 17, 2021
#9
+2111
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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
+121000
+1

OK, catmg   !!!

CPhill  Mar 17, 2021
#11
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This was an unnecessarily difficult problem, but thank you very much for the continuous assistance :)

owentout  Mar 17, 2021
#12
+121000
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They're  all  unnecessarily  difficult....until find out that many aren't    (LOL!!!)

CPhill  Mar 17, 2021