Sum and Difference Identities

Step 1: Express 75° as a sum of known angles: 75° = 45° + 30°

Step 2: Use the sine sum formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

Step 3: Substitute A = 45° and B = 30°: sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°)

Step 4: Substitute known values: sin(45°) = √2/2 cos(45°) = √2/2 sin(30°) = 1/2 cos(30°) = √3/2

Step 5: Calculate: sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6/4) + (√2/4) = (√6 + √2)/4

Answer: sin(75°) = (√6 + √2)/4

Step 1: Express 15° as a difference of known angles: 15° = 45° - 30°

Step 2: Use the cosine difference formula: cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Step 3: Substitute A = 45° and B = 30°: cos(15°) = cos(45°)cos(30°) + sin(45°)sin(30°)

Step 4: Substitute known values: cos(45°) = √2/2 sin(45°) = √2/2 cos(30°) = √3/2 sin(30°) = 1/2

Step 5: Calculate: cos(15°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6/4) + (√2/4) = (√6 + √2)/4

Answer: cos(15°) = (√6 + √2)/4

Step 1: Find cos(α) using Pythagorean identity: sin²(α) + cos²(α) = 1 (3/5)² + cos²(α) = 1 9/25 + cos²(α) = 1 cos²(α) = 16/25 cos(α) = 4/5 (positive in Quadrant I)

Step 2: Find sin(β) using Pythagorean identity: sin²(β) + cos²(β) = 1 sin²(β) + (5/13)² = 1 sin²(β) + 25/169 = 1 sin²(β) = 144/169 sin(β) = 12/13 (positive in Quadrant I)

Step 3: Use sine sum formula: sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

Step 4: Substitute values: sin(α + β) = (3/5)(5/13) + (4/5)(12/13) = 15/65 + 48/65 = 63/65

Answer: sin(α + β) = 63/65

Step 1: Recognize this pattern: This matches the cosine sum formula

Step 2: Recall the cosine sum formula: cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Step 3: Compare to our expression: cos(x)cos(y) - sin(x)sin(y)

Step 4: This is exactly cos(x + y): cos(x)cos(y) - sin(x)sin(y) = cos(x + y)

Answer: cos(x + y)

Step 1: Use tangent sum formula: tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B))

Step 2: Substitute A = π/4 and B = x: tan(π/4 + x) = (tan(π/4) + tan(x))/(1 - tan(π/4)tan(x))

Step 3: Use tan(π/4) = 1: tan(π/4 + x) = (1 + tan(x))/(1 - (1)tan(x))

Step 4: Simplify: tan(π/4 + x) = (1 + tan(x))/(1 - tan(x))

Step 5: Verification complete: Left side = Right side ✓

This identity is useful for shifting tangent functions.

Answer: Proven ✓