Double Angle Formula for Cosine Last Updated : 23 Jul, 2025 Comments Improve Suggest changes Like Article Like Report Double angle formula for cosine is a trigonometric identity that expresses cos(2θ) in terms of cos(θ) and sin(θ) the double angle formula for cosine is, cos 2θ = cos2θ - sin2θ. The formula is particularly useful in simplifying trigonometric expressions and solving equations involving trigonometric functions. In this article, we will learn about, Double Angle Formula for Cosine, Double Angle Formula for Cosine Derivation, Double Angle Formula for Cosine Examples and others. Table of Content Cos Double Angle FormulasDerivation of Double Angle Formula for CosineSample Problems on Double Angle Formulas for CosineFAQs on Double Angle Formula for CosineCos Double Angle FormulasIn trigonometry, cos 2x is a double-angle identity. Because the cos function is a reciprocal of the secant function, it may also be represented as cos 2x = 1/sec 2x. It's a significant trigonometric identity that may be used for a variety of trigonometric and integration problems. The value of cos 2x repeats after every π radians, cos 2x = cos (2x + π). It has a considerably narrower graph than cos x. It's a trigonometric function that returns the cos function value of a double angle. Cos Double Angle FormulasThe cos double angle formula can be simplified further by using sine cosine identity and that is: cos 2x = cos2 x - sin2 x Putting sin2 x = 1 - cos2 x, the formula becomes, cos 2x = cos2 x - (1 - cos2 x) cos 2x = 2 cos2 x - 1 Putting cos2 x = 1 - sin2 x, the formula becomes, cos 2x = (1 - sin2 x) - sin2 x cos 2x = 1 - 2 sin2 x Derivation of Double Angle Formula for CosineFormula for cos 2x can be derived by using the sum angle formula for cosine function. We already know, cos (A + B) = cos A cos B - sin A sin B To calculate the value of cosine double angle, the angle A must be equal to angle B. Putting A = B we get, cos (A + A) = cos A cos A - sin A sin A cos 2A = cos2 A - sin2 A This derives the formula for double angle of cosine ratio. Read More, Trigonometric IdentitiesTrigonometry TableHalf Angle FormulaeSample Problems on Double Angle Formulas for CosineProblem 1. If cos x = 3/5, find the value of cos 2x using the formula. Solution: We have, cos x = 3/5 Clearly, sin x = 4/5 Using the formula we get, cos 2x = cos2 x - sin2 x = (3/5)2 - (4/5)2 = 9/25 - 16/25 = -7/25 Problem 2. If cos x = 12/13, find the value of cos 2x using the formula. Solution: We have, cos x = 12/13 Clearly, sin x = 5/13 Using the formula we get, cos 2x = cos2 x - sin2 x = (12/13)2 - (5/13)2 = 144/169 - 25/169 = 119/169 Problem 3. If sin x = 3/5, find the value of cos 2x using the formula. Solution: We have, sin x = 3/5 Clearly cos x = 4/5 Using the formula we get, cos 2x = cos2 x - sin2 x = (4/5)2 - (3/5)2 = 16/25 - 9/25 = 7/25 Problem 4. If tan x = 12/5, find the value of cos 2x using the formula. Solution: We have, tan x = 12/5 Clearly sin x = 12/13 and cos x = 5/13 Using the formula we get, cos 2x = cos2 x - sin2 x = (5/13)2 - (12/13)2 = 25/169 - 144/169 = -119/169 Problem 5. If sec x = 17/8, find the value of cos 2x using the formula. Solution: We have, sec x = 17/8 Clearly cos x = 8/17 and sin x = 15/17 Using the formula we get, cos 2x = cos2 x - sin2 x = (8/17)2 - (15/17)2 = 64/289 - 225/289 = -161/289 Problem 6. If cot x = 15/8, find the value of cos 2x using the formula. Solution: We have, cot x = 15/8 Clearly cos x = 15/17 and sin x = 8/17 Using the formula we get, cos 2x = cos2 x - sin2 x = (15/17)2 - (8/17)2 = 225/289 - 64/289 = 161/289 Problem 7. If cos2 x = 5/8, find the value of cos 2x using the formula. Solution: We have, cos2 x = 5/8 Using the formula we get, cos 2x = 2 cos2 x - 1 = 2 (5/8) - 1 = 5/4 - 1 = 1/4 Problem 8. If sin2 x = 6/7, find the value of cos 2x using the formula. Solution: We have, sin2 x = 6/7 Using the formula we get, cos 2x = 1 - 2 sin2 x = 1 - 2 (6/7) = 1 - 12/7 = -5/7 Comment More infoAdvertise with us Next Article What are Numbers? G gurjotloveparmar Follow Improve Article Tags : Mathematics School Learning Maths MAQ Maths-Formulas Similar Reads Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields 5 min read Basic ArithmeticWhat are Numbers?Numbers are symbols we use to count, measure, and describe things. 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Practising these questions with solutions may ensure that students can do a good practice of all types of questions that can be framed in the examination. This ensures that they excel in their final 13 min read RD Sharma Class 12 Solutions for MathsRD Sharma Solutions for class 12 provide solutions to a wide range of questions with a varying difficulty level. With the help of numerous sums and examples, it helps the student to understand and clear the chapter thoroughly. Solving the given questions inside each chapter of RD Sharma will allow t 13 min read Like