Reduce cos x − sin x to the Sine and Cosine of a Single Expression

Solution

We use the standard form:

\[ a\cos x+b\sin x=R\cos(x+\alpha) \]

where

\[ R=\sqrt{a^2+b^2} \]

Given expression:

\[ \cos x-\sin x \]

Here,

\[ a=1, \qquad b=-1 \]

Now,

\[ R=\sqrt{1^2+(-1)^2} \]

\[ =\sqrt{1+1} \]

\[ =\sqrt{2} \]

Let

\[ \cos x-\sin x = \sqrt{2}\cos(x+\alpha) \]

Using the identity:

\[ \sqrt{2}\cos(x+\alpha) = \sqrt{2}(\cos x\cos\alpha-\sin x\sin\alpha) \]

Comparing coefficients,

\[ \sqrt{2}\cos\alpha=1 \]

\[ \cos\alpha=\frac{1}{\sqrt{2}} \]

and

\[ \sqrt{2}\sin\alpha=1 \]

\[ \sin\alpha=\frac{1}{\sqrt{2}} \]

Therefore,

\[ \alpha=\frac{\pi}{4} \]

Hence,

\[ \boxed{ \cos x-\sin x = \sqrt{2}\cos\left(x+\frac{\pi}{4}\right) } \]

Also, using sine form:

\[ \boxed{ \cos x-\sin x = \sqrt{2}\sin\left(\frac{\pi}{4}-x\right) } \]

Final Answer

\[ \boxed{ \cos x-\sin x = \sqrt{2}\cos\left(x+\frac{\pi}{4}\right) } \]

or

\[ \boxed{ \cos x-\sin x = \sqrt{2}\sin\left(\frac{\pi}{4}-x\right) } \]