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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. The expression is 2cot⁡(x)−2cot⁡(x)cos⁡2(x) 2 \cot(x) - 2 \cot(x) \cos^2(x) .
2. We will use trigonometric identities to simplify the expression.
3. Graphing technology will be used to visualize the expression.
4. The goal is to find an equivalent expression and prove its equivalence.

STEP 2

1. Graph the expression using technology.
2. Simplify the trigonometric expression.
3. Prove the equivalence of the simplified expression to the original.

STEP 3

Use a graphing calculator or software (such as Desmos, GeoGebra, or a graphing calculator app) to graph the expression 2cot⁡(x)−2cot⁡(x)cos⁡2(x) 2 \cot(x) - 2 \cot(x) \cos^2(x) .
- Input the expression into the graphing tool. - Observe the graph for 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 sin⁡2(x)=1−cos⁡2(x) \sin^2(x) = 1 - \cos^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:
2⋅cos⁡(x)sin⁡(x)⋅sin⁡2(x) 2 \cdot \frac{\cos(x)}{\sin(x)} \cdot \sin^2(x)
- Simplify the expression:
2cos⁡(x)sin⁡(x) 2 \cos(x) \sin(x)

STEP 5

Prove that the simplified expression is equivalent to the original expression.
- Start with the simplified expression:
2cos⁡(x)sin⁡(x) 2 \cos(x) \sin(x)
- Recall the double angle identity for sine: sin⁡(2x)=2sin⁡(x)cos⁡(x) \sin(2x) = 2 \sin(x) \cos(x) .
- Therefore, the simplified expression can be written as:
sin⁡(2x) \sin(2x)
- Compare this with the original expression 2cot⁡(x)−2cot⁡(x)cos⁡2(x) 2 \cot(x) - 2 \cot(x) \cos^2(x) .
- We have shown that both simplify to sin⁡(2x) \sin(2x) , proving their equivalence.

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