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

  1. #1
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    cos(x)-sin(x) as single trigonometric function

    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.
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  2. #2
    Senior Member eumyang's Avatar
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    Multiply the expression by sqrt of 2 and it's reciprocal (which is really multiplying by 1):
    cos x - sin x \\<br />
 = \sqrt{2} \cdot \frac{1}{\sqrt{2}} \left( cos x - sin x \right)

    Distribute just the fraction (1 over sqrt 2):
    = \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):
    = \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:
    = \sqrt{2} cos \left( x + \frac{\pi}{4} \right)
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