SOLUTION: verifyin trigonometric identities. (1/1-sinx)+(1/1+sinx)=2sec^2x

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!
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

The LCD in the LHS is the product of the two denominators. Since they are binomial conjugates, the product is the difference of two squares.

Apply the Pythagorean Identity to the LHS denominator and simplify the LHS numerator

Recall that

Q.E.D.

John

My calculator said it, I believe it, that settles it

RELATED QUESTIONS
1/sinx-1 - 1/sinx+1 =... (answered by Alan3354)
Verify trigonometric identities... (answered by lwsshak3)
Solve the following trigonometric identities.1.sinx... (answered by Alan3354)
Could you please show me how to prove the following identities... A) cos2x=2cos^2x-1 B) (answered by lwsshak3)
How would I verify that the following (below) is an identity? (1/sinx -1) - (1/sinx... (answered by Gogonati)
prove the following identities: sin^2x+4sinx+3/cos^^2x=... (answered by stanbon)
VERIFY each trigonometric identity: I'd appreciate any help:) a)... (answered by stanbon)
Verify the following trig identities 1) sin^2x/cosx=secx-cosx 2) 1+sinx/1-sinx -... (answered by lwsshak3)
1/sinx (answered by ewatrrr)