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Question: How do you simplify \(\cos (\alpha + \beta ) - \cos (\alpha - \beta )\) ?...

How do you simplify cos⁡(α+β)−cos⁡(α−β)\cos (\alpha + \beta ) - \cos (\alpha - \beta ) ?

Explanation

Solution

Trigonometry is that branch of mathematics that tells us the relation between the sides of a right-angled triangle that is the base, the perpendicular and the hypotenuse. All the trigonometric functions can be converted into any other trigonometric function through some identities and have great importance in mathematics and real-life too. To find the trigonometric value of large angles, there are many ways. In the given question, we will find the cosine of the sum and the difference of α\alpha and β\beta by using the sum or difference identity.

Complete step by step answer:
We have to simplify cos⁡(α+β)−cos⁡(α−β)\cos (\alpha + \beta ) - \cos (\alpha - \beta )
We know that
⇒cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\Rightarrow \cos (\alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
⇒cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\Rightarrow \cos (\alpha - \beta ) = \cos \alpha \cos \beta + \sin \alpha \sin \beta

Using these values in the given equation, we get –
⇒cos⁡(α+β)−cos⁡(α−β)=cos⁡αcos⁡β−sin⁡αsin⁡β−(cos⁡αcos⁡β+sin⁡αsin⁡β)\Rightarrow \cos (\alpha + \beta ) - \cos (\alpha - \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta - (\cos \alpha \cos \beta + \sin \alpha \sin \beta )
We know that when −(a+b)=−a−b - (a + b) = - a - b
⇒cos⁡(α+β)−cos⁡(α−β)=cos⁡αcos⁡β−sin⁡αsin⁡β−cos⁡αcos⁡β−sin⁡αsin⁡β\Rightarrow \cos (\alpha + \beta ) - \cos (\alpha - \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta - \cos \alpha \cos \beta - \sin \alpha \sin \beta
Writing the like terms together we get –
⇒cos⁡(α+β)−cos⁡(α−β)=cos⁡αcos⁡β−cos⁡αcos⁡β−sin⁡αsin⁡β−sin⁡αsin⁡β ⇒cos⁡(α+β)−cos⁡(α−β)=−2sin⁡αsin⁡β  \Rightarrow \cos (\alpha + \beta ) - \cos (\alpha - \beta ) = \cos \alpha \cos \beta - \cos \alpha \cos \beta - \sin \alpha \sin \beta - \sin \alpha \sin \beta \\\ \Rightarrow \cos (\alpha + \beta ) - \cos (\alpha - \beta ) = - 2\sin \alpha \sin \beta \\\
Hence, the simplified form of cos⁡(α+β)−cos⁡(α−β)\cos (\alpha + \beta ) - \cos (\alpha - \beta ) is −2sin⁡αsin⁡β- 2\sin \alpha \sin \beta.

Note: Trigonometric functions are of six types that are sine, cosine, tangent, cosecant, secant and cotangent. Sine, cosine and tangent are the main functions while cosecant, secant and cotangent are their reciprocals respectively. According to the sum identity of cosines, the cosine of the sum of any two angles a and b is equal to the difference of the product of the cosine of one angle (a) and the cosine of the other angle (b) and the product of the sine of angle a and sine of angle b, that is, cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos (a + b) = \cos a\cos b - \sin a\sin b, similarly the cosine of the difference of any two angles a and b is equal to the sum of the product of the cosine of one angle (a) and the cosine of the other angle (b) and the product of the sine of angle a and sine of angle b, that is, cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos (a - b) = \cos a\cos b + \sin a\sin b. Using these two identities we can solve similar questions.