How to use trig identities to transform one side of the equation into the other?

For the first one, expand the binomials on the left side.

You can use the FOIL method, or whatever method you like best. (I'll use A for theta)

(1 + cosA)(1 - cosA) = sin^2 A

1 - cosA + cosA - cos^2 A = sin^2 A

1 - cos^2 A = sin^2 A

Using the pythagorean identity, we know that 1 - cos^2 A = sin^2 A. Therefore, the identity is proven.

For the second one, the pythagorean theorem is used again. Since we know that sin^2 A + cos^2 A = 1, we can insert it into the equation where there is a 1, which is on the right side.

sin^2 A + cos^2 A = 2sin^2 A - (sin^2 A + cos^2 A)

Now you just simplify the right side by adding and subtracting like terms

sin^2 A + cos^2 A = 2sin^2 A - sin^2 A - cos^2 A

sin^2 A + cos^2 A = sin^2 A - cos^2 A

Left side equals right side, therefore identity is true.

tan O = sin O / cos O cot O = a million / tan O cot O = cos O / sin O tan O / cot O = (sin O / cos O) / (cos O / sin O) cancel out sin O in numerator and denominator = (a million / cos O) / (cos O / a million) cancel out cos O in numerator and denominator = (a million / a million) / (a million / a million) = a million / a million = a million