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When you are not sure what to do when solving these equations (or proving identities), try rewriting any sec's, csc's, tan's or cot's in terms of sin's and/or cos's. (Note: Don't start doing this automatically every time. Just do it when you see no other way.) Rewriting our equation this way we get:
Next, we'll eliminate the fractions by multiplying both sides by the lowest common denominator (LCD). The LCD of cos(A) and 2 is 2cos(A).
On the left side we must use the Distributive Property:
Each denominator cancels with some part of 2cos(A):
This is a quadratic. So we want a zero on one side as we solve it. I'm going to subtract the entire left side from both sides (since I like the squared term to have a positive coefficient):
Now we factor:
From the Zero Product Property: or
Solving these for cos(A) we get: or
We should recognize that a cos is never equal to -2. So there is no solution for that equation. So only
cos(A) = 1/2 is true.
Normally we would proceed to find A at this point. But the problem asks for the value of sec(A). Since sec(A) is the reciprocal of cos:
(