9. Use the difference Identity for cosine to expand and simplify the given expression cos(x-pi/2)

9. Use the difference Identity for cosine to expand and simplify the given expression cos(x-pi/2)

10. 4cos^3x - 3cosx = 0 0 is less than or equal to X < 2pi

11. find the exact value for sec 5pi/12

9.

cos(x - π/2) = cos(x) cos(π/2) + sin(x) sin(π/2)
. . . . . . . . . . = cos(x) * 0 + sin(x) * 1
. . . . . . . . . . = sin(x)

10.

4 cos³x - 3 cosx = 0
cosx (4 cos²x - 3) = 0

cosx = 0 -----> x = π/2, 3π/2
cos²x = 3/4
cosx = ± √3/2 ----> x = π/6, 5π/6, 7π/6, 11π/6

11.

cos(5π/12) = cos(π/4 + π/6)
. . . . . . . . . = cos(π/4) cos(π/6) - sin(π/4) sin(π/6)
. . . . . . . . . = (√2/2) (√3/2) - (√2/2) (1/2)
. . . . . . . . . = √6/4 - √2/4
. . . . . . . . . = (√6 - √2) / 4

sec(5π/12) = 1/cos(5π/12)
. . . . . . . . . = 4/(√6-√2) * (√6+√2)/(√6+√2)
. . . . . . . . . = 4(√6+√2)/(6-2)
. . . . . . . . . = √6 + √2

The cosine difference identity goes as follows:
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

So in order to evaluate cos(x - pi/2), we apply the identity.

cos(x - pi/2) = cos(x)cos(pi/2) + sin(x)sin(pi/2)
= cos(x)(0) + sin(x)(1)
= sin(x)

11. To find the exact value of 5pi/12, you must express it as the sum or difference of two known unit circle values. Since we have the difference identity handy, we will express it as a difference.

5pi/12 is the same as 8pi/12 - 3pi/12 which is the same as 2pi/3 - pi/4.
Note that 2pi/3 and pi/4 are two known unit circle values. This means

sec(5pi/12) = 1/cos(5pi/12)
= 1/cos(2pi/3 - pi/4)
= 1/[cos(2pi/3)cos(pi/4) + sin(2pi/3)sin(pi/4)]
= 1/[sqrt(3)/2 * sqrt(2)/2 + sqrt(3)/2 sqrt(2)/2 ]
= 1/[sqrt(6)/4 + sqrt(6)/4]
= 1/[2sqrt(6)/4]
= 1/[sqrt(6)/2]
= 2/sqrt(6)
= 2sqrt(6)/6
= sqrt(6)/3