SOLUTION: what is value sin(x) in terms of sec(x)?

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Question 903431: what is value sin(x) in terms of sec(x)?
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Express sin(x) in terms of sec(x).
Start with this identity:

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

replace cos%28x%29 by 1%2F%28sec%28x%29%29:

sin%5E2%28x%29%2B%281%2F%28sec%28x%29%29%29%5E2=1

sin%5E2%28x%29%2B1%2F%28sec%5E2%28x%29%29=1

Multiply through by sec˛(x):

sec%5E2%28x%29sin%5E2%28x%29%2B1=sec%5E2%28x%29

Subtract 1 from both sides:

sec%5E2%28x%29sin%5E2%28x%29=sec%5E2%28x%29-1

Divide both sides by sec˛(x):

sin%5E2%28x%29=%28sec%5E2%28x%29-1%29%2F%28sec%5E2%28x%29%29

Take ± square root:

sin%28x%29%22%22=%22%22%22%22%2B-sqrt%28%28sec%5E2%28x%29-1%29%2F%28sec%5E2%28x%29%29%29

Simplify:

sin%28x%29%22%22=%22%22%22%22%2B-sqrt%28sec%5E2%28x%29-1%29%2F%28sec%28x%29%29

[Which sign to take will depend on the quadrant x is in.]

Edwin