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. :)))
I hate to be this person but this problem has to be solved whilst keeping -tan alone on the right side
oh noessss. :(((((
Welp, at least I tried.
Luckily, it looks like CPhill is typing so hopefully we can learn from him. :))
I would very much hope so! Thank you for your gracious attempt though!
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 )
-1 / cot x =
-1 / ( 1 /tan x)
-1 (tan x) =
cot x cot x cot^2 x
____ * _______ = _______
1 cot x cot x
cot ^2x csc^2 x
_____ - _______ =
cot x cot x
(cot^2 x -csc^2 x)
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. :)))
This was an unnecessarily difficult problem, but thank you very much for the continuous assistance :)