Sum and difference formulas are trigonometric identities used to find the values of angles by expressing them as the sum or difference of known standard angles, such as 30°, 45°, and 60°, and are useful for solving problems, simplifying expressions, and proving identities.
Solved Problems
Problem 1: Prove the identity sin(2A) = 2 sin A cos A
Solution:
Using the sum formula for sine: sin(A + A) = sin A cos A + cos A sin A
sin(2A) = sin(A + A) = sin A cos A + sin A cos A
= 2 sin A cos A
Problem 2: Prove the identity cos(2A) = cos² A - sin² A
Solution:
Using the sum formula for cosine: cos(A + A) = cos A cos A - sin A sin A
cos(2A)
= cos(A + A) = cos A cos A - sin A sin A
= cos² A - sin² A
Problem 3: Simplify sin(x + 45°)
Solution:
Using the sum formula: sin(x + 45°) = sin x cos 45° + cos x sin 45°
Substitute known values: cos 45° = sin 45° = √2/2
= sin(x + 45°)
= sin x (√2/2) + cos x (√2/2)
= (√2/2)(sin x + cos x)
Problem 4: Simplify cos(x - 30°)
Solution:
Use the difference formula: cos(x - 30°) = cos x cos 30° + sin x sin 30°
Substitute known values: cos 30° = √3/2, sin 30° = 1/2
= cos(x - 30°)
= cos x cos 30° + sin x sin 30°
= cos x (√3/2) + sin x (1/2)
Problem 5: Calculate cos 15°
Solution:
15° = 45° − 30°
Using identity:
cos(A − B) = cosA cosB + sinA sinB
cos 15° = cos(45° − 30°)
= cos45° cos30° + sin45° sin30°Substitute values:
cos45° = √2/2
sin45° = √2/2
cos30° = √3/2
sin30° = 1/2
cos 15° = (√2/2)(√3/2) + (√2/2)(1/2)= √6/4 + √2/4
= (√6 + √2) / 4
Problem 6: Calculate tan 105°
Solution:
105° = 45° + 60°
Using identity:
tan(A + B) = (tanA + tanB) / (1 − tanA tanB)So,
tan 105° = tan(45° + 60°)
= (tan45° + tan60°) / (1 − tan45° tan60°)Substitute values:
tan45° = 1
tan60° = √3tan 105° = (1 + √3) / (1 − √3)
Rationalize denominator:
tan 105° = (1 + √3)(1 + √3) / [(1 − √3)(1 + √3)]
= (1 + 2√3 + 3) / (1 − 3)
= (4 + 2√3) / (−2)
= −2 − √3
tan 105° = −(2 + √3)
Practice Questions
Q1. Simplify: sin(x + 45°)
Q2. Calculate cos(60° - 15°)
Q3. Evaluate tan(135°).
Q4. Solve for x: sin(x + 30°) = √3/2
Q5. Prove the identity: sin(A + B) = sin A cos B + cos A sin B
Q6. Prove the identity: sin(A + B) sin(A - B) = sin²A - sin²B
Q7. Prove the identity: cos(A - B) = cos A cos B + sin A sin B
Q8. Simplify the following expression as much as possible: (sin(x+y) + sin(x-y))(cos(x+y) - cos(x-y))
Q9. Prove the following triple angle formula using sum and difference identities: sin(3x) = 3 sin(x) - 4 sin³(x)
Q10. Prove the identity: tan(A + B) = (tan A + tan B) / (1 - tan A tan B)