NDA Maths Trigonometry: Important Formulas and Questions for 2026 Exam

NDA Maths Trigonometry: Mastering trigonometry is crucial for success in competitive exams like NDA, where time management and accuracy are key. Here we highlight Rapid Result (RR) Methods, a set of powerful shortcuts and specific identities aimed at simplifying complex trigonometric problems. By understanding essential concepts like A+B+C=180°, X+1/X=2 forms, and transformation identities, candidates can quickly solve problems with precision. 

These methods are designed to save valuable time during exams, enabling candidates to focus on more challenging questions and improve their overall performance. Whether you're new to trigonometry or aiming to sharpen your skills, these techniques are indispensable for efficient problem-solving.

Introduction to Rapid Result (RR) Methods

The Rapid Result Method (RR Method) is a powerful approach designed to solve trigonometry problems swiftly. It involves recognizing specific patterns and applying pre-derived identities. 

Trigonometric Identities for A + B + C = π (180°)

These identities are important results that candidates should imprint in their minds and affix to their study tables, as questions have appeared and will continue to appear in competitive exams.

• Rule 1: If A + B + C = π (or 180°), then the identity tan A + tan B + tan C = tan A tan B tan C holds true.

• Example: If A + B + C = 180°, find the value of (tan A + tan B + tan C) / (tan A tan B tan C).

• Solution: Using the identity, the expression simplifies to (tan A tan B tan C) / (tan A tan B tan C) = 1.

• Rule 2: If A + B + C = 180° (e.g., A, B, C are angles of a triangle), then the identity cos 2A + cos 2B + cos 2C = -1 - 4 cos A cos B cos C holds true.

• Example: If A, B, C form a triangle and cos 2A + cos 2B + cos 2C = -1, determine the condition for A, B, C.

• Solution: Given cos 2A + cos 2B + cos 2C = -1.

• Substituting the identity: -1 - 4 cos A cos B cos C = -1.

• This implies -4 cos A cos B cos C = 0.

• Therefore, cos A cos B cos C = 0.

Identities of the Form X + 1/X = 2

If X + 1/X = 2, then it implies X = 1. Consequently, for any power 'n', the expression Xⁿ + 1/Xⁿ will also be equal to 2.

• Case 1: If cos θ + sec θ = 2, then (cos θ)ⁿ + (sec θ)ⁿ = 2 for any integer 'n'.

• Case 2: If tan θ + cot θ = 2, then (tan θ)ⁿ + (cot θ)ⁿ = 2 for any integer 'n'.

Key Identity: (1 + tan α)(1 + tan β) = 2

This is a most important RR Method. If the result (1 + tan α)(1 + tan β) = 2 is observed, then it implies that the sum of the angles α + β will always be 45°.

• Rule: If (1 + tan α)(1 + tan β) = 2, then α + β = 45°.

• Example: If (1 + tan θ)(1 + tan 9θ) = 2, find the value of tan(10θ).

• Solution: Using this RR Method, θ + 9θ = 45°.

• This means 10θ = 45°.

• Therefore, tan(10θ) = tan(45°) = 1.

Reciprocal Trigonometric Pair Identities

These rules describe reciprocal relationships between trigonometric functions, useful for solving equations involving sum/difference of reciprocal pairs.

• Rule 1: If sec θ + tan θ = m, then sec θ - tan θ = 1/m.

• Rule 2: If cosec θ + cot θ = m, then cosec θ - cot θ = 1/m.

• Example: If sec θ + tan θ = 4, find the value of cos θ.

• Solution:

1. From Rule 1, sec θ - tan θ = 1/4.

2. Adding the two equations: (sec θ + tan θ) + (sec θ - tan θ) = 4 + 1/4.

3. 2 sec θ = 16/4 + 1/4 = 17/4.

4. sec θ = 17/8.

5. Therefore, cos θ = 8/17.

Pythagorean Triplets with a sin θ + b cos θ = c

If you have an equation a sin θ + b cos θ = c and the coefficients a, b, c form a Pythagorean Triplet (meaning c² = a² + b²), then you can directly determine the trigonometric ratios.

• Rule:

• sin θ = a/c

• cos θ = b/c

• tan θ = a/b

• Example: If 7 sin θ + 24 cos θ = 25, find the value of sin θ + cos θ.

• Solution: Observe that 7² + 24² = 49 + 576 = 625 = 25².

• Thus, (7, 24, 25) form a Pythagorean Triplet.

• Applying the rule directly: sin θ = 7/25 and cos θ = 24/25.

• Therefore, sin θ + cos θ = 7/25 + 24/25 = 31/25.

Product Identities (60-Degree Family)

These are very important RR Methods for simplifying products of trigonometric functions.

• Rule 1: sin θ sin(60° - θ) sin(60° + θ) = (1/4)sin(3θ)

• Rule 2: cos θ cos(60° - θ) cos(60° + θ) = (1/4)cos(3θ)

• Rule 3: tan θ tan(60° - θ) tan(60° + θ) = tan(3θ)

• Example: Find the value of cos 20° cos 40° cos 80°.

• Solution: Applying Rule 2 by setting θ = 20° (which implies 60°-θ = 40°, and 60°+θ = 80°):

• Value = (1/4)cos(3 * 20°) = (1/4)cos(60°) = (1/4) * (1/2) = 1/8.

Transformation Identity for a sin x + b cos x = c

• Rule: If a sin x + b cos x = c, then the value of b sin x - a cos x = ±√(a² + b² - c²).

• Example: If 3 sin θ + 5 cos θ = 5, find the value of 5 sin θ - 3 cos θ.

• Solution: Here, a=3, b=5, c=5.

• Applying the rule: 5 sin θ - 3 cos θ = ±√(3² + 5² - 5²) = ±√(9 + 25 - 25) = ±√9 = ±3.

Complementary Angle Identities (θ1 + θ2 = 90°)

This is very important for simplifying expressions involving products of tangent and cotangent functions.

• Rule 1: If θ1 + θ2 = 90°, then tan θ1 * tan θ2 = 1.

• Rule 2: If θ1 + θ2 = 90°, then cot θ1 * cot θ2 = 1.

• Example: Find the value of tan 15° tan 25° tan 45° tan 65° tan 75°.

• Solution:

• (tan 15° * tan 75°) = 1 (since 15° + 75° = 90°)

• (tan 25° * tan 65°) = 1 (since 25° + 65° = 90°)

• tan 45° = 1

• Therefore, the entire product is 1 * 1 * 1 = 1.

Average Angle Shortcut for Sums of Sines/Cosines

This is very important for simplifying certain trigonometric series.

• Rule: For a series of angles θ1, θ2, …, θn, if an expression involving sums of sines and cosines is given, its value can sometimes be directly calculated as tan( (θ1 + θ2 + … + θn) / n ).

• Example: For angles 20°, 50°, 40°, 70°, an expression implicitly involving sums of sines and cosines is given. Determine its value.

• Solution:

• Sum of angles = 20° + 50° + 40° + 70° = 180°.

• Number of angles (n) = 4.

• Average angle = 180° / 4 = 45°.

• The value of the expression is tan(45°) = 1.

NDA Maths Trigonometry Important Questions

Here are 8 important NDA Maths Trigonometry questions:

1. Prove that sin(A + B + C) = 0 where A + B + C = 180°.

2. If tan A = 1, find the value of tan(45° + A).

3. Solve for θ if 2sin θ + √3 = 0, 0° ≤ θ ≤ 360°.

4. Prove the identity sin² θ + cos² θ = 1 using Pythagorean theorem.

5. Find the value of sin 60° · cos 30°.

6. If sin A = 3/5, find the value of cos A.

7. Simplify (1 + tan² θ) / sec² θ.

8. If tan θ = 3/4, find the values of sin θ and cos θ.