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For cos θ ≠ 0, prove that cosec 2θ + cot 2θ = cot θ Explain why cot θ ≠ cosec 2θ + cot 2θ when cos θ = 0 - AQA - A-Level Maths Pure - Question 9 - 2020 - Paper 3
Question 9
For cos θ ≠ 0, prove that cosec 2θ + cot 2θ = cot θ Explain why cot θ ≠ cosec 2θ + cot 2θ when cos θ = 0
Worked Solution & Example Answer:For cos θ ≠ 0, prove that cosec 2θ + cot 2θ = cot θ Explain why cot θ ≠ cosec 2θ + cot 2θ when cos θ = 0 - AQA - A-Level Maths Pure - Question 9 - 2020 - Paper 3
Step 1
Prove that cosec 2θ + cot 2θ = cot θ
Answer
To prove the identity, we start with the left-hand side:
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Substituting Definitions:
- We know that:
[ \text{cosec } 2 heta = \frac{1}{\sin 2\theta} ] and
[ \text{cot } 2\theta = \frac{\cos 2\theta}{\sin 2\theta} ] - Thus,
[ \text{cosec } 2\theta + \text{cot } 2\theta = \frac{1 + \cos 2\theta}{\sin 2\theta} ]
- We know that:
-
Using Trigonometric Identities:
- Next, we apply:
[ \cos 2\theta = 2\cos^2 \theta - 1 ] and
[ \sin 2\theta = 2\sin \theta \cos \theta ] - Hence, substituting these into our equation yields: [ \frac{1 + (2\cos^2 \theta - 1)}{2\sin \theta \cos \theta} = \frac{2\cos^2 \theta}{2\sin \theta \cos \theta} = \frac{\cos^2 \theta}{\sin \theta \cos \theta} = \cot \theta ]
- Next, we apply:
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Final Verification:
- Therefore, we have shown that: [ \text{cosec } 2\theta + \text{cot } 2\theta = \cot \theta ]
Step 2
Explain why cot θ ≠ cosec 2θ + cot 2θ when cos θ = 0
Answer
When ( \cos \theta = 0 ), it corresponds to angles like ( \theta = 90° + k180° ), leading to:
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Undefined Terms:
- In this case, ( \text{cot } \theta = \frac{\cos \theta}{\sin \theta} ) is undefined because ( \cos \theta = 0 ).
- Similarly, both ( \text{cosec } 2\theta ) and ( \text{cot } 2\theta ) are undefined due to ( \sin 2\theta = 0 ).
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Conclusion:
- Hence, since both sides of the equation become undefined under these conditions, it stands that: [ \text{cot } \theta \neq \text{cosec } 2\theta + \text{cot } 2\theta ] when ( \cos \theta = 0 ).