Question

Use the sum and difference identities to rewrite the following expression as a trigonometric function of a single number.cos(15") cos(15) – sin(15") sin(15)

Answer 1

From the sum and difference identities expressed as

`\begin{gathered} \sin (A+B)=\sin A\cos B+\sin B\cos A\text{ ---- equation 1} \\ \sin (A-B)=\sin A\cos B-\sin B\cos A\text{ ----- equation 2} \\ \text{cos}(A+B)=\cos A\cos B-\sin A\sin B\text{ ------ equation 3} \\ \text{cos}(A-B)=\cos A\cos B+\sin A\sin B\text{ ------ equation }4 \end{gathered}`

Given:

`\cos (15)\cos (15)-\sin (15)\sin (15)`

The above expression satisfies equation 3, where

`\begin{gathered} A=15, \\ B=15 \end{gathered}`

Thus,

`\begin{gathered} \cos (15)\cos (15)-\sin (15)\sin (15)=\cos (15+15) \\ =\cos (30) \end{gathered}`

Hence, the expression is simplified to be

`\cos (30)`