Final answer:
To prove the trigonometric identity, we'll use the given hint: cos(90°) = sin(0°) and cos²(0) + sin²(0) = 1. Using the angle subtraction formula for cosine, we rewrite the left-hand side of the identity and then simplify to reach the right-hand side of the identity.
Step-by-step explanation:
To prove the trigonometric identity, we'll use the given hint: cos(90°) = sin(0°) and cos²(0) + sin²(0) = 1. Let's start with the left-hand side of the identity: cos(x - π/3) + sin(π/6 - x).
Using the angle subtraction formula for cosine: cos(a - b) = cos(a)cos(b) + sin(a)sin(b), we can rewrite the left-hand side as:
cos(x)cos(π/3) + sin(x)sin(π/6)
Since cos(π/3) = 1/2 and sin(π/6) = 1/2, we have:
(1/2)cos(x) + (1/2)sin(x)
Combine the two terms using the identity sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2):
2(1/2)sin((x + π/6)/2)cos((x - π/6)/2)
Now, simplify the terms inside the sin and cos functions:
sin((x + π/6)/2) = sin(x/2 + π/12) = sin(x/2)cos(π/12) + cos(x/2)sin(π/12)
cos((x - π/6)/2) = cos(x/2 - π/12) = cos(x/2)cos(π/12) - sin(x/2)sin(π/12)
Substitute these values back into the equation:
2(1/2)[sin(x/2)cos(π/12) + cos(x/2)sin(π/12)][cos(x/2)cos(π/12) - sin(x/2)sin(π/12)]
Expand the equation:
(sin(x/2)cos(π/12)cos(x/2) - sin(x/2)sin(π/12)sin(x/2)) + (cos(x/2)sin(π/12)cos(x/2) - cos(x/2)cos(π/12)sin(x/2))
Simplify the terms:
(1/2)sin²(x/2) + (1/2)cos²(x/2) = sin²(x/2) + cos²(x/2) = 1
The simplified equation is equal to the right-hand side of the identity, which is cos(x). Therefore, we have proven the trigonometric identity: cos(x - π/3) + sin(π/6 - x) = cos(x).