Math  /  Trigonometry

Question5. Given the expression 2cot(x)2cot(x)cos2(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
Prove that the simplified expression sin(2x) \sin(2x) is equal to the original expression 2cot(x)2cot(x)cos2(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)sin2(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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