SOLUTION: Given that x = sec^2θ and y+2 = cot^2θ, find y in terms of x.

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Question 1196469: Given that x = sec^2θ and y+2 = cot^2θ, find y in terms of x.
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

If x+=+sec%5E2%28theta%29 then 1%2Fx+=+1%2F%28sec%5E2%28theta%29%29+=+cos%5E2%28theta%29

Check out this list of trig identities
https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf
Look under the "pythagorean identities" subsection, on page 2, to find the identity 1+%2B+cot%5E2%28theta%29+=+csc%5E2%28theta%29
This rearranges to cot%5E2%28theta%29+=+csc%5E2%28theta%29-1

So we could say:
y%2B2+=+cot%5E2%28theta%29

y%2B2+=+csc%5E2%28theta%29-1

y+=+csc%5E2%28theta%29-1-2

y+=+csc%5E2%28theta%29-3

y+=+1%2F%28sin%5E2%28theta%29%29-3

y+=+1%2F%281-cos%5E2%28theta%29%29-3

y+=+1%2F%281-1%2Fx%29-3

y+=+1%2F%28x%2Fx-1%2Fx%29-3

y+=+1%2F%28%28x-1%29%2Fx%29-3

y+=+x%2F%28x-1%29+-+3

We could optionally continue like so
y+=+x%2F%28x-1%29+-+3

y+=+x%2F%28x-1%29+-+3%28x-1%29%2F%28x-1%29

y+=+%28x+-+3%28x-1%29%29%2F%28x-1%29

y+=+%28x+-+3x%2B3%29%2F%28x-1%29

y+=+%28-2x%2B3%29%2F%28x-1%29