SOLUTION: How does one prove the following identity: 9sec^2θ - 5tan^2θ = 5 + 4sec^2θ Thank you very much for your help.

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Question 1192730: How does one prove the following identity:
9sec^2θ - 5tan^2θ = 5 + 4sec^2θ
Thank you very much for your help.
Found 3 solutions by Alan3354, Edwin McCravy, MathTherapy:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
How does one prove the following identity:
9sec^2θ - 5tan^2θ = 5 + 4sec^2θ
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5sec^2 = 5tan^2 + 5
sec^2 = tan^2 + 1
QED
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If you want, multiply thru by cos^2
---> 1 = sin^2 + cos^2 the Pythagorean identity.
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PS I wouldn't call 9sec^2θ - 5tan^2θ = 5 + 4sec^2θ an identity.
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Edwin, I don't consider that to be an identity.

Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
Alan, you aren't allowed to work with but one side of an identity
at a time.

9sec%5E2%28theta%29+-+5tan%5E2%28theta%29%22%22=%22%225+%2B+4sec%5E2%28theta%29

Work ONLY with the left side

9sec%5E2%28theta%29+-+5tan%5E2%28theta%29

Use the identity 1+%2Btan%5E2%28theta%29=sec%5E2%28theta%29 solved for tan2(θ),
which is tan%5E2%28theta%29=sec%5E2%28theta%29-1 to replace tan%5E2%28theta%29:

9sec%5E2%28theta%29+-+5%28sec%5E2%28theta%29-1%5E%22%22%5E%22%22%29

Distribute the -5 to remove the big parentheses:

9sec%5E2%28theta%29-5sec%5E2%28theta%29%2B5

4sec%5E2%28theta%29%2B5

5%2B4sec%5E2%28theta%29

Now that is exactly like the right side. So the identity is proved.

Edwin

Answer by MathTherapy(10557) About Me  (Show Source):
You can put this solution on YOUR website!
How does one prove the following identity:
9sec^2θ - 5tan^2θ = 5 + 4sec^2θ
Thank you very much for your help.
PROVING RIGHT-SIDE
9 sec2 θ  -  5 tan2 θ = 5 + 4 sec2 θ
9 sec2 θ  -  5 (sec2 θ - 1) = 5 + 4 sec2 θ -------- Substituting sec2 θ  -  1 for tan2 θ
9 sec2 θ  -  5 sec2 θ + 5 = 5 + 4 sec2 θ
4 sec2 θ + 5 = 5 + 4 sec2 θ  <======== PROVEN