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9. (a) Prove that tanθ + cotθ = 2cosec2θ, θ ≠ nπ/2, n ∈ ℤ (b) Hence explain why the equation tanθ + cotθ = 1 does not have any real solutions. - Edexcel - A-Level Maths Pure - Question 11 - 2017 - Paper 1
Question 11
9. (a) Prove that tanθ + cotθ = 2cosec2θ, θ ≠ nπ/2, n ∈ ℤ (b) Hence explain why the equation tanθ + cotθ = 1 does not have any real solutions.
Worked Solution & Example Answer:9. (a) Prove that tanθ + cotθ = 2cosec2θ, θ ≠ nπ/2, n ∈ ℤ (b) Hence explain why the equation tanθ + cotθ = 1 does not have any real solutions. - Edexcel - A-Level Maths Pure - Question 11 - 2017 - Paper 1
Step 1
Prove that tanθ + cotθ = 2cosec2θ
Answer
To prove the equation, we start with the left-hand side:
oveline{tanθ + cotθ} = rac{sinθ}{cosθ} + rac{cosθ}{sinθ} = rac{sin^2θ + cos^2θ}{sinθcosθ} = rac{1}{sinθcosθ}$$ Using the double angle formula, we know that: $$sin2θ = 2sinθcosθ$$ Thus, we can rewrite the left-hand side as: $$ rac{1}{ rac{1}{2}sin2θ} = rac{2}{sin2θ} = 2cosec2θ$$ This shows that: $$tanθ + cotθ = 2cosec2θ$$Step 2
Explain why tanθ + cotθ = 1 does not have any real solutions
Answer
Starting from the equation:
we can substitute: rac{sinθ}{cosθ} + rac{cosθ}{sinθ} = 1
Multiplying through by sinθcosθ gives:
Since we know that , we can set up the inequality:
This leads to:
However, at the maximum value of sin2θ, it reaches 1, showing that:
This means there are no values for theta where this holds true, confirming that the equation has no real solutions as: .