Sum and difference identities in trigonometry are essential tools for simplifying and solving trigonometric equations. These identities relate the trigonometric functions of the sum or difference of two angles to the functions of the individual angles. Understanding and applying these identities can simplify complex trigonometric expressions and solve equations more efficiently.

In this article on Trigonometric identities, we will learn about what are Suma and Difference Trigonometric Identities, their related practice problem and others in detail.

What are Trigonometric identities?

Trigonometric identities refer to equations that contain the values of the trigonometric functions, which are valid for all values of the variables involved. Such identities should express relations between different trigonometric functions and angles; therefore, they allow a simplification and manipulation of trigonometric expressions. They include basic identities like sin²θ + cos²θ = 1, Pythagoras identities, sum and difference formulas, double-angle formulas, half-angle formulas, etc.

Sum and Difference Identities

In trigonometry, sum and difference identities are useful for simplifying expressions involving the sine, cosine, and tangent functions of sums or differences of angles. Here are the key identities:

Sum Identities

• Sine Sum Identity: sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
• Cosine Sum Identity: cos(a + b) = cos(a)cos(b) – sin(a)sin(b)
• Tangent Sum Identity: tan(a + b) = f{tan(a) + tan(b)}/{1 – tan(a)tan(b)}

Difference Identities

• Sine Difference Identity: sin(a – b) = sin(a)cos(b) – cos(a)sin(b)
• Cosine Difference Identity: cos(a – b) = cos(a)cos(b) + sin(a)sin(b)
• Tangent Difference Identity: [tan(a – b) = {tan(a) – tan(b)}/{1 + tan(a)tan(b)}

Sum and Difference Identities Practice 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 sin 75°

Solution:

Express as a sum: 75° = 45° + 30°

Use the sum formula: sin 75° = sin(45° +30° ) = sin 45° cos 30° + cos 45° sin 30°

Substitute known values: sin 45° = cos 45° = √2/2, cos 30° = √3/2, sin 30° = 1/2

= sin 75°

= sin 45° cos 30° + cos 45° sin 30°

= (√2/2)(√3/2) + (√2/2)(1/2)

Now, Simplify = sin 75° = (√6 + √2) / 4

Problem 7: Calculate cos 15°

Solution:

We shall express it as a difference: 15° = 45° – 30°

Use the difference formula: cos 15° = cos( 45° – 30°) = cos 45° cos 30° + sin 45° sin 30°

Substitute known values: cos 45° = sin 45° = √2/2, cos 30° = √3/2, sin 30° = 1/2

= cos 15°

= cos 45° cos 30° + sin 45° sin 30°

= (√2/2)(√3/2) + (√2/2)(1/2)

Now we just need to simplify: cos 15° = (√6 + √2) / 4

Problem 7: Calculate tan 105°

Solution:

We shall express this as a sum: 105° = 45° + 60°

Using the sum formula: tan 105° = tan( 45° + 60°) = (tan 45° + tan 60°) / (1 – tan 45° tan 60°)

Now, we substitute known values: tan 45° = 1, tan 60° = √3

= tan 105°

= (tan 45° + tan 60°) / (1 – tan 45° tan 60°)

= (1 + √3) / (1 – 1(√3))

Simplifying tan 105° = -(1 + √3)

Sum and Difference Identities Worksheet

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) = 3sin(x) – 4sin³(x)

Q10. Prove the identity: tan(A + B) = (tan A + tan B) / (1 – tan A tan B)

Read More,

• List of All trigonometric Identities
• Sum and Difference Formula Identities in trigonometry
• Trigonometric Identities
• Inverse trigonometric Identities

FAQs on Sum and difference identities

What are Basic Sum and Difference Identities for Sine and Cosine?

The basic Sum and Difference identities for sine and cosine are:

Sum Identities:

• sin(A + B) = sin A cos B + cos A sin B
• cos(A + B) = cos A cos B – sin A sin B

Difference Identities:

• sin(A – B) = sin A cos B – cos A sin B
• cos(A – B) = cos A cos B + sin A sin B

How can Sum and Difference Identities be used to prove Double Angle Formulas?

Sum and Difference identities can be used to derive double angle formulas by setting A = B in the Sum identities. For example:

• cos(2A) = cos(A + A) = cos A cos A – sin A sin A = cos²A – sin²A
• sin(2A) = sin(A + A) = sin A cos A + cos A sin A = 2 sin A cos A
• tan(2A) = tan(A + A) = (tan A + tan A) / (1 – tan A tan A) = 2 tan A / (1 – tan²A)

Whatis Relationship between Sum and Difference Identities and Product-to-Sum Formulas?

• sin A cos B = 1/2[sin(A + B) + sin(A – B)]
• cos A cos B = 1/2[cos(A + B) + cos(A – B)]
• sin A sin B = 1/2[cos(A – B) – cos(A + B)]