cos(x)-sin(x) as single trigonometric function

Jun 2010
11
0
Can anybody help me with this?

Show that cos(x) - sin(x) can be expressed in terms of a single trigonometric function. Then plot its graph.
 
Jan 2010
278
138
Multiply the expression by sqrt of 2 and it's reciprocal (which is really multiplying by 1):
\(\displaystyle cos x - sin x \\
= \sqrt{2} \cdot \frac{1}{\sqrt{2}} \left( cos x - sin x \right)\)

Distribute just the fraction (1 over sqrt 2):
\(\displaystyle = \sqrt{2} \left( (cos x)\left(\frac{1}{\sqrt{2}}\right) - (sin x)\left(\frac{1}{\sqrt{2}}\right) \right)\)

I know that cos (pi/4) = sin (pi/4) = 1/sqrt(2):
\(\displaystyle = \sqrt{2} \left( (cos x)\left(cos \frac{\pi}{4}\right) - (sin x)\left(sin \frac{\pi}{4}\right) \right)\)

Use the cosine of a sum identity:
\(\displaystyle = \sqrt{2} cos \left( x + \frac{\pi}{4} \right)\)
 
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