SOLUTION: if cos theta= 3x, then tan^2 theta=?

Algebra ->  Trigonometry-basics -> SOLUTION: if cos theta= 3x, then tan^2 theta=?      Log On


   

Question 631840: if cos theta= 3x, then tan^2 theta=?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
There's probably more than one way to solve this. Perhaps the easiest is based on recognizing that cos and tan are both connected to sec:
sec%28theta%29+=+1%2Fcos%28theta%29
and
tan%5E2%28theta%29%2B1+=+sec%5E2%28theta%29
From these two equations we have a "path" to go from cos to tan^2

You've been given that cos%28theta%29+=+3x. We can convert this to sec by finding its reciprocal (1/cos):
sec%28theta%29+=+1%2F%283x%29
Now we can use the other identity by squaring sec:
sec%5E2%28theta%29+=+%281%2F%283x%29%29%5E2+=+1%2F9x%5E2
And then inserting that into the identity:
tan%5E2%28theta%29+%2B++1+=+1%2F9x%5E2
For just tan^2, we subtract 1 from each side:
tan%5E2%28theta%29+=+%281%2F9x%5E2%29+-+1
This should be an acceptable answer. If not, then get common denominators and subtract:
tan%5E2%28theta%29+=+%281%2F9x%5E2%29+-+%289x%5E2%2F9x%5E2%29+=+%281-9x%5E2%29%2F9x%5E2