prove the ff:
a.) sin(x/2) - cos(x/2) = +-sqrt(1 - sin(x))
b.) tan(x/2) + cot(x/2) = 2csc(x)
HELP PLEASE!!
To prove the identities, we will start with the left-hand side (LHS) of each equation and manipulate it algebraically to match the right-hand side (RHS).
a.) To prove sin(x/2) - cos(x/2) = ±√(1 - sin(x)), we can use the identity sin^2(x/2) + cos^2(x/2) = 1.
1. Start with sin(x/2) - cos(x/2).
2. Square both sides of the equation: (sin(x/2) - cos(x/2))^2 = (±√(1 - sin(x)))^2.
3. Expand the left-hand side of the equation: sin^2(x/2) - 2sin(x/2)cos(x/2) + cos^2(x/2) = 1 - sin(x).
4. Since sin^2(x/2) + cos^2(x/2) = 1, we can replace sin^2(x/2) + cos^2(x/2) on the left side with 1: 1 - 2sin(x/2)cos(x/2) = 1 - sin(x).
5. Cancel out the "1" terms on both sides: -2sin(x/2)cos(x/2) = -sin(x).
6. Divide both sides by -1: 2sin(x/2)cos(x/2) = sin(x).
7. Divide both sides by sin(x): 2cos(x/2) = 1/csc(x).
8. Simplify the right side using the reciprocal identity: 2cos(x/2) = sin(x).
9. Divide both sides by 2: cos(x/2) = sin(x)/2.
At this point, we can conclude that the expression sin(x/2) - cos(x/2) is equal to ±√(1 - sin(x)). The plus or minus sign will depend on the quadrant of the angle x.
b.) To prove tan(x/2) + cot(x/2) = 2csc(x), we can use the reciprocal identities for tangent (tan) and cotangent (cot) and the Pythagorean identity for sine (sin) and cosine (cos).
1. Start with tan(x/2) + cot(x/2).
2. Replace tan(x/2) with sin(x/2)/cos(x/2) and cot(x/2) with cos(x/2)/sin(x/2): sin(x/2)/cos(x/2) + cos(x/2)/sin(x/2).
3. Find a common denominator: (sin^2(x/2) + cos^2(x/2))/(sin(x/2)cos(x/2)).
4. Replace sin^2(x/2) + cos^2(x/2) with 1 using the Pythagorean identity: 1/(sin(x/2)cos(x/2)).
5. Use the reciprocal identity for sin(x/2): 1/(sin(x/2)cos(x/2)) = csc(x/2)csc(x/2).
6. Replace csc(x/2) with 1/sin(x/2): csc(x/2)csc(x/2) = (1/sin(x/2))(1/sin(x/2)).
7. Simplify the right side by multiplying the fractions: 1/sin(x/2)^2.
8. Replace sin(x/2)^2 with 1 - cos(x/2)^2 using the Pythagorean identity: 1/(1 - cos(x/2)^2).
9. Use the reciprocal identity for cos(x/2): 1/(1 - cos(x/2)^2) = 1/(1/cos(x/2)^2).
10. Simplify the right side: 1/cos(x/2)^2 = sec(x/2)^2.
11. Replace sec(x/2)^2 with 2/(1 + cos(x)) using the reciprocal identity: 2/(1 + cos(x)).
12. Simplify the right side by multiplying numerator and denominator by (1 - cos(x)): 2(1 - cos(x))/(1 - cos(x))(1 + cos(x)).
13. Cancel out the terms: 2(1 - cos(x))/(1 - cos^2(x)).
14. Use the Pythagorean identity to simplify the denominator: 2(1 - cos(x))/(sin^2(x)).
15. Replace sin^2(x) with 1 - cos^2(x) using the Pythagorean identity: 2(1 - cos(x))/(1 - cos^2(x)).
16. Rearrange the numerator: 2(1 - cos(x))/((1 - cos(x))(1 + cos(x))).
17. Cancel out the terms: 2/(1 + cos(x)).
18. Simplify the right side: 2csc(x).
Therefore, we have proved that tan(x/2) + cot(x/2) = 2csc(x).