Prove the following identities:
13. tan(x) + sec(x) = (cos(x)) / (1-sin(x))
*Sorry for any confusing parenthesis.*
My work:
I simplified the left side to
a. ((sinx) / (cosx)) + (1 / cosx) , then
b. (sinx + 1) / cosx = (cos(x)) / (1-sin(x))
I don't know how to verify the rest.
good start
LS = sinx/cosx + 1/cosx
= (sinx + 1)/cosx
multiply top and bottom by (sinx - 1)/(sinx - 1)
= (sinx + 1)/cosx * (sinx - 1)/(sinx - 1)
= (sin^2 x - 1)/(cosx(sinx - 1))
= cos^2 x/(cosx(sinx - 1))
= cosx/(sinx - 1)
= RS
To verify the rest of the identity, you can cross-multiply and manipulate the expression until you obtain an equality.
Starting with:
(sin(x) + 1) / cos(x) = (cos(x)) / (1 - sin(x))
Cross-multiplying, we have:
(sin(x) + 1)(1 - sin(x)) = (cos(x))(cos(x))
Expanding and simplifying the left side, we get:
(1 - sin(x) + sin(x) - sin^2(x)) = cos^2(x)
Combining like terms, we have:
1 - sin^2(x) = cos^2(x)
Using the Pythagorean Identity sin^2(x) + cos^2(x) = 1, we know that:
1 - sin^2(x) = cos^2(x)
Therefore, the left and right sides of the equation are equal, thus proving the identity.