Math Doubts

Cosine angle difference identity


$(1).\,\,$ $\cos{(a-b)}$ $\,=\,$ $\cos{a}\cos{b}$ $+$ $\sin{a}\sin{b}$

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


Let $a$ and $b$ be two variables, which are used to represent two angles in this case. The subtraction of angle $b$ from angle $a$ is the difference between them, and it is written as $a-b$, which is a compound angle. The cosine of a compound angle $a$ minus $b$ is written as $\cos{(a-b)}$ in trigonometry.

cos angle difference identity

The cosine of subtraction of angle $b$ from angle $a$ is equal to the sum of the products of cosines of angles $a$ and $b$, and sines of angles $a$ and $b$.

$\cos{(a-b)}$ $\,=\,$ $\cos{a} \times \cos{b}$ $+$ $\sin{a} \times \sin{b}$

This mathematical equation can be called the cosine angle difference trigonometric identity in mathematics.


The cosine angle difference identity is possibly used in two different cases in trigonometry.


The cosine of difference of two angles is expanded as the sum of the products of cosines of angles and sines of angles.

$\implies$ $\cos{(a-b)}$ $\,=\,$ $\cos{(a)}\cos{(b)}$ $+$ $\sin{(a)}\sin{(b)}$


The sum of the products of cosines of angles and sines of angles is simplified as the cosine of difference of two angles.

$\implies$ $\cos{(a)}\cos{(b)}$ $+$ $\sin{(a)}\sin{(b)}$ $\,=\,$ $\cos{(a-b)}$


The angle difference identity in cosine function is written in several forms but the following three forms are some popularly used forms in the world.

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

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

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


Learn how to derive the cosine of angle difference trigonometric identity by a geometric method in trigonometry.

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