Question 592883: verifyin trigonometric identities.
(1/1-sinx)+(1/1+sinx)=2sec^2x Found 2 solutions by jsmallt9, solver91311:Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website! These identities can be hard because there is no "recipe" that one can memorize and then apply to each one.
When I looked at your problem here is what I saw:
The right side is expressed in terms of sec(x)
Since sec(x) is the reciprocal of cos(x) it might be helpful to find a way to express the left side in terms of cos(x)
The left side is expressed in terms of sin(x). One way to convert sin's to cos's is to use sin^2(x) + cos^2(x) = 1 or cos^2(x) = 1 - sin^2(x)
The sin's on the left are not squared. But from the factoring pattern I know I can get 1 - sin^2(x) in both denominators by multiplying each fraction by appropriate expressions (as you'll see shortly). This will turn both denominators into cos^2(x)! And not only that, since the denominators will then be the same, the fractions can then be added!!
Adding the fractions will turn the left side into a single term (which is good because the right side is also a single term).
So let's put these ideas into action:
Create the denominators:
which simplifies to:
The denominators can be replaced by cos^2(x):
And we can add the fractions together. The sin's in the numerator cancel out giving us:
We are very close now. Factoring out 2 on the left (after all, 2 is a factor on the right):
And the fraction on the left can be replaced by sec^2(x):
And we're done!
(
