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QuestionSimplify 1cos⁡x−cos⁡x=\frac{1}{\cos x} - \cos x =

Studdy Solution

STEP 1

Assumptions1. The variable x is a real number. . The cosine function, denoted as cos(x), is a function that takes a real number x (an angle in radians) and returns the cosine of that angle.

STEP 2

The given expression is a difference of two terms 1cos⁡x\frac{1}{\cos x} and cos⁡x\cos x. To simplify such expressions, we often look for common denominators.

STEP 3

To find a common denominator, we can rewrite cos⁡x\cos x as cos⁡x1\frac{\cos x}{1}.
1cos⁡x−cos⁡x1\frac{1}{\cos x}-\frac{\cos x}{1}

STEP 4

The common denominator of these two fractions is cos⁡x\cos x. We can rewrite each fraction with cos⁡x\cos x as the denominator.
1cos⁡x−cos⁡x⋅cos⁡xcos⁡x\frac{1}{\cos x}-\frac{\cos x \cdot \cos x}{\cos x}

STEP 5

implify the second fraction.
1cos⁡x−cos⁡2xcos⁡x\frac{1}{\cos x}-\frac{\cos^2 x}{\cos x}

STEP 6

Now that the two fractions have the same denominator, we can combine them into a single fraction.
1−cos⁡2xcos⁡x\frac{1-\cos^2 x}{\cos x}

STEP 7

Recognize that 1−cos⁡2x1-\cos^2 x is a Pythagorean identity and can be replaced with sin⁡2x\sin^2 x.
sin⁡2xcos⁡x\frac{\sin^2 x}{\cos x}

STEP 8

The expression is now simplified. The final simplified form of the given expression is sin⁡2xcos⁡x\frac{\sin^2 x}{\cos x}.

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