1) Use the cofunction identites to evaluate the expression without the aid of a calculator.
cos^2 18 degrees + cos^2 72 degrees.
Error
The cofunction identities that are useful here are
cos(90-x) = sinx and
sin(90-x) = cosx
and the fact that cos(A-B) = cosAcosB + sinAsinB
so cos^2 72 = (cos72)(cos72) = (sin18)(sin18)
then your
cos^2 18 degrees + cos^2 72 degrees
= cos18cos18 + sin18sin18
= cos(18-18)
= cos 0
= 1
To evaluate the expression without a calculator, we can utilize the cofunction identities. The cofunction identities state that the cosine of an angle is equal to the sine of its complement and vice versa. In other words:
sin θ = cos (90° - θ)
cos θ = sin (90° - θ)
Now, let's break down the given expression:
cos^2 18 degrees + cos^2 72 degrees
Using the identity cos θ = sin (90° - θ), we can rewrite cos^2 18 degrees as sin^2 (90° - 18 degrees):
sin^2 (90° - 18 degrees) + cos^2 72 degrees
Next, let's evaluate the complementary angle for 18 degrees:
90° - 18 degrees = 72 degrees
Now, the expression becomes:
sin^2 72 degrees + cos^2 72 degrees
Since sin^2 θ + cos^2 θ = 1 (a well-known trigonometric identity), we can simplify the expression further:
1 + 1 = 2
Therefore, without using a calculator, the value of cos^2 18 degrees + cos^2 72 degrees is 2.