a)Express as a function of cos (theta)
2 sin^2(theta) - 1
b)Express as a function of sin (theta) or cos (theta)
tan^2(theta) - 2 sec(theta)sin(theta)
a) To express 2 sin^2(theta) - 1 as a function of cos(theta), we can use the trigonometric identity:
sin^2(theta) + cos^2(theta) = 1
We can rearrange this equation to solve for sin^2(theta):
sin^2(theta) = 1 - cos^2(theta)
Now substitute this expression into the original equation:
2 sin^2(theta) - 1 = 2(1 - cos^2(theta)) - 1
Simplifying further:
= 2 - 2cos^2(theta) - 1
= 1 - 2cos^2(theta)
Therefore, 2 sin^2(theta) - 1 can be expressed as a function of cos(theta) as 1 - 2cos^2(theta).
b) To express tan^2(theta) - 2 sec(theta)sin(theta) as a function of sin(theta) or cos(theta), we can use the following trigonometric identities:
tan(theta) = sin(theta)/cos(theta) and sec(theta) = 1/cos(theta)
Substituting these identities into the original equation:
tan^2(theta) - 2 sec(theta)sin(theta) = (sin(theta)/cos(theta))^2 - 2 (1/cos(theta))sin(theta)
= sin^2(theta)/cos^2(theta) - 2 sin(theta)/cos(theta)
Multiplying both terms by cos^2(theta) to eliminate the division:
= (sin^2(theta) - 2 sin(theta)cos(theta))/cos^2(theta)
Using the trigonometric identity sin(2theta) = 2sin(theta)cos(theta), we can simplify further:
= sin^2(theta) - 2 sin(theta)cos(theta)
Therefore, tan^2(theta) - 2 sec(theta)sin(theta) can be expressed as a function of sin(theta) or cos(theta) as sin^2(theta) - 2 sin(theta)cos(theta).