Math Doubts

Angle difference formulas

A trigonometric identity to expand a trigonometric function having difference of two angles is called the angle difference identity. In trigonometry, there are four angle difference trigonometric identities and they’re used as formulas in mathematics. Let’s start to study all the angle difference identities with proofs.

Sine angle difference formula

$(1) \,\,\,\,$ $\sin{(A-B)}$ $\,=\,$ $\sin{A}\cos{B}$ $-$ $\cos{A}\sin{B}$

$(2) \,\,\,\,$ $\sin{(x-y)}$ $\,=\,$ $\sin{x}\cos{y}$ $-$ $\cos{x}\sin{y}$

$(3) \,\,\,\,$ $\sin{(\alpha-\beta)}$ $\,=\,$ $\sin{\alpha}\cos{\beta}$ $-$ $\cos{\alpha}\sin{\beta}$

Cosine angle difference formula

$(1) \,\,\,\,$ $\cos{(A-B)}$ $\,=\,$ $\cos{A}\cos{B}$ $+$ $\sin{A}\sin{B}$

$(2) \,\,\,\,$ $\cos{(x-y)}$ $\,=\,$ $\cos{x}\cos{y}$ $+$ $\sin{x}\sin{y}$

$(3) \,\,\,\,$ $\cos{(\alpha-\beta)}$ $\,=\,$ $\cos{\alpha}\cos{\beta}$ $+$ $\sin{\alpha}\sin{\beta}$

Tangent angle difference formula

$(1) \,\,\,\,$ $\tan{(A-B)}$ $\,=\,$ $\dfrac{\tan{A}-\tan{B}}{1+\tan{A}\tan{B}}$

$(2) \,\,\,\,$ $\tan{(x-y)}$ $\,=\,$ $\dfrac{\tan{x}-\tan{y}}{1+\tan{x}\tan{y}}$

$(3) \,\,\,\,$ $\tan{(\alpha-\beta)}$ $\,=\,$ $\dfrac{\tan{\alpha}-\tan{\beta}}{1+\tan{\alpha}\tan{\beta}}$

Cotangent angle difference formula

$(1) \,\,\,\,$ $\cot{(A-B)}$ $\,=\,$ $\dfrac{\cot{B}\cot{A}+1}{\cot{B}-\cot{A}}$

$(2) \,\,\,\,$ $\cot{(x-y)}$ $\,=\,$ $\dfrac{\cot{y}\cot{x}+1}{\cot{y}-\cot{x}}$

$(3) \,\,\,\,$ $\cot{(\alpha-\beta)}$ $\,=\,$ $\dfrac{\cot{\beta}\cot{\alpha}+1}{\cot{\beta}-\cot{\alpha}}$

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved