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Math  /  Trigonometry

Question5. Given the expression 2cot⁡(x)−2cot⁡(x)cos⁡2(x)2 \cot (x)-2 \cot (x) \cos ^{2}(x), a. Use technology to graph the expression [3 marks] b. Determine an equivalent trigonometric expression [2 marks] c. Then prove that your expression is equal to the given expression. [3 marks]

Studdy Solution

STEP 1

1. We are given a trigonometric expression and need to graph it, find an equivalent expression, and prove the equivalence.
2. Technology such as graphing calculators or software can be used for graphing.
3. Trigonometric identities will be useful in simplifying the expression.

STEP 2

1. Graph the expression using technology.
2. Simplify the expression to find an equivalent trigonometric expression.
3. Prove that the simplified expression is equivalent to the original expression.

STEP 3

Use a graphing tool or software (such as Desmos, GeoGebra, or a graphing calculator) to graph the expression 2cot⁡(x)−2cot⁡(x)cos⁡2(x) 2 \cot(x) - 2 \cot(x) \cos^2(x) .
- Enter the expression into the graphing tool. - Observe the graph and note any key features such as intercepts, asymptotes, and periodicity.

STEP 4

Simplify the expression 2cot⁡(x)−2cot⁡(x)cos⁡2(x) 2 \cot(x) - 2 \cot(x) \cos^2(x) .
- Factor out the common term 2cot⁡(x) 2 \cot(x) :
2cot⁡(x)(1−cos⁡2(x)) 2 \cot(x) (1 - \cos^2(x))
- Use the Pythagorean identity 1−cos⁡2(x)=sin⁡2(x) 1 - \cos^2(x) = \sin^2(x) :
2cot⁡(x)sin⁡2(x) 2 \cot(x) \sin^2(x)
- Recall that cot⁡(x)=cos⁡(x)sin⁡(x) \cot(x) = \frac{\cos(x)}{\sin(x)} , so:
2cos⁡(x)sin⁡(x)sin⁡2(x) 2 \frac{\cos(x)}{\sin(x)} \sin^2(x)
- Simplify the expression:
2cos⁡(x)sin⁡(x) 2 \cos(x) \sin(x)
- Recognize that sin⁡(2x)=2sin⁡(x)cos⁡(x) \sin(2x) = 2 \sin(x) \cos(x) , so the expression is:
sin⁡(2x) \sin(2x)
The equivalent trigonometric expression is sin⁡(2x) \sin(2x) .

STEP 5

Prove that the simplified expression sin⁡(2x) \sin(2x) is equal to the original expression 2cot⁡(x)−2cot⁡(x)cos⁡2(x) 2 \cot(x) - 2 \cot(x) \cos^2(x) .
- Start with the simplified expression:
sin⁡(2x)=2sin⁡(x)cos⁡(x) \sin(2x) = 2 \sin(x) \cos(x)
- Substitute back to verify:
2cos⁡(x)sin⁡(x)sin⁡2(x)=2cos⁡(x)sin⁡(x) 2 \frac{\cos(x)}{\sin(x)} \sin^2(x) = 2 \cos(x) \sin(x)
- Simplify:
2cos⁡(x)sin⁡(x)=2cos⁡(x)sin⁡(x) 2 \cos(x) \sin(x) = 2 \cos(x) \sin(x)
Both sides of the equation are equal, thus proving the equivalence.
The equivalent trigonometric expression is sin⁡(2x) \sin(2x) , and it is proven to be equal to the original expression.

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