Power Reduction Formulae
Now Trig Frog will examine the Power Reduction Formulae:
sin^2 A = (1 - cos2A)/2
cos^2 A = (1 + cos2A)/2
tan^2 A = (1 - cos2A)/(1 + cos2A)
These formulae can be proven using the Double Angle Formulae. Observe:
cos2A = 1 - 2sin^2 A Subtract -1.
-1 + cos2A = -2sin^2 A Divide by -2
Answer: sin^2 A = (1 - cos2A)/2
cos2A = 2cos^2 A - 1 Add 1.
1 + cos2A = 2cos^2 A Divide by 2.
Answer: cos^2 A = (1+cos2A)/2
tan^2 A = (sin^2 A)/(cos^2 A) Replace sin^2A with the power reduction formula.
tan^2 A = ((1 - cos2A)/2)/(cos^2 A) Replace cos^2A with the power reduction formula.
Answer: tan^2 A = (1 - cos2A)/(1 + cos2A)
Example
Write sin^4 x as a sum of the first powers of the cosines of multiple angles.
Solution
Trig Frog suggest that thou use the power reduction formlae as much as possible!
sin^4 x = (sin^2 x)^2 Use the rules of exponents to separate the exponent.
= ((1 - cos2x)/2)^2 Use the power reduction formula.
= 1/4( 1 - 2cos2x + cos^2 2x) Expand the binomial to get rid of the outer ^2.
= 1/4(1 - 2cos2x + ((1 + cos4x)/2)) Use the power reducing formula.
= 1/4 - 1/2cos2x + 1/8 + 1/8cos4x Take out the 1/2..
= 3/8 - 1/2cos2x + 1/8cos4x Perform the operations and simplify the problem.
= 1/8(3 - 4cos2x + cos4x) Factor out the -1/2 and 1/8.
