SOLUTION: If tan²x = 1 + 2tan²y Show that cos2x + sin²y = 0

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Question 1161703: If tan²x = 1 + 2tan²y
Show that
cos2x + sin²y = 0
Answer by Edwin McCravy(20081) About Me  (Show Source):
You can put this solution on YOUR website!
If tan²x = 1 + 2tan²y
Show that
cos2x + sin²y = 0
tan²x = 1 + 2tan²y

Use the identity

          1%2Btan%5E2%28theta%29=sec%5E2%28theta%29 

rewritten as 

          tan%5E2%28theta%29=sec%5E2%28theta%29-1

sec%5E2%28x%29-1+=+1+%2B+2%28sec%5E2%28y%29-1%29
sec%5E2%28x%29-1+=+1%2B2sec%5E2%28y%29-2
sec%5E2%28x%29-1+=+2sec%5E2%28y%29-1
sec%5E2%28x%29=2sec%5E2%28y%29
1%2Fcos%5E2%28x%29=2%281%2Fcos%5E2%28y%29%29
1%2Fcos%5E2%28x%29=2%2Fcos%5E2%28y%29
cos%5E2%28y%29=2cos%5E2%28x%29

Use the identity

          sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1 

rewritten as 

          cos%5E2%28theta%29=1-sin%5E2%28theta%29

on the left side

Use the identity 

          cos%282theta%29=2cos%5E2%28theta%29-1

rewritten as

          2cos%5E2%28theta%29=cos%282theta%29%2B1

on the right side

1-sin%5E2%28y%29=cos%282x%29%2B1

Subtract 1 from both sides:

-sin%5E2%28y%29=cos%282x%29

add sin²(x) to both sides:

0=cos%282x%29%2Bsin%5E2%28x%29

Swap sides:

cos%282x%29%2Bsin%5E2%28x%29=0 

Edwin