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Calculus/Trig Identities

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Expert: - 1/10/2010


Question
Prove the indentity:

1/1-SINx + 1/1+SINx = 2 SEC^2x

COT^2x + SEC^2x -TAN^2 = CSC^2



Note: X stands for theta
Thank You SO much if you are able to help!

Answer
1/(1-sonx) + 1/(1+sinx) should have a common denominator.
The common denominator is 1-sin²x.
Multiply the 1st by (1+sinx)/(1+sinx) and the 2nd by (1-sinx)/(1-sinx).
This gives (1-sinx)/(1-sin²x) + (1+sinx)/(1-sin²x).
They can be combined, so that the numerator reduces to 2.
The denominator follows the rule that sin²x + cos²x = 1 and become cos²x.
It is known that 2/cos²x = 2sec²x.

The function cot²x + sec²x - tan²x can be converted to sinx and cosx.
It is cos²x/sin²x + 1/cos²x - sin²x/cos²x.
The 2nd and 3rd functions can be combined, giving cos²x/sin²x + (1-sin²x)/cos²x.
Since sin²x + cos²x = 1, the numerator is cos²x.  From this, the 2nd fraction is 1.
To combine with the first fraction, the number needs to be multiplied by sin²x/sin²x.
This gives cos²x/sin²x + sin²x/sin²x = (cos²x + sin²x)/sin²x.
Noting that the numerator can be seen to be 1, we have 1/sin²x, which is known as csc²x.

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