In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 15 - 2020 - Paper 1
Question 15
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
(a) Show that
cosec θ − sin θ ≡ ... show full transcript
Worked Solution & Example Answer:In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 15 - 2020 - Paper 1
Step 1
Show that cosec θ − sin θ ≡ cos θ cot θ
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Answer
To show that cosec θ − sin θ ≡ cos θ cot θ, we start from the left-hand side:
Rewrite cosec θ in terms of sin θ:
cosecθ=sinθ1
Therefore,
cosecθ−sinθ=sinθ1−sinθ
Combine the terms under a common denominator:
cosecθ−sinθ=sinθ1−sin2θ
Use the Pythagorean identity, where:
1−sin2θ=cos2θ
So,
cosecθ−sinθ=sinθcos2θ
This can be factored as:
sinθcos2θ=cosθ⋅sinθcosθ=cosθcotθ
Thus, we have shown that:
cosecθ−sinθ≡cosθcotθ
Finally, state that θ ≠ (180n)° for any integer n, as this condition is necessary to ensure that the expressions are defined.
Step 2
Hence, or otherwise, solve for 0 < x < 180°
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Answer
Starting from:
cosecx−sinx=cosx⋅cot(3x−50°)
Rewrite cosec x in terms of sin x:
sinx1−sinx=cosx⋅sin(3x−50°)cos(3x−50°)
Combine the left-hand side:
sinx1−sin2x=cosx⋅cot(3x−50°)
Using the identity, we have:
sinxcos2x=cosx⋅cot(3x−50°)
Cancel cos x (assuming cos x ≠ 0):
sinxcosx=cot(3x−50°
Set equations equal:
cotx=cot(3x−50°)
Implying:
x=3x−50°+kimes180°,k∈Z
Solve for x:
a) First solution:
x=3x−50°⟹2x=50°⟹x=25°
b) Second solution:
x=3x−50°+180°⟹2x=230°⟹x=115°
Check that both solutions fall within the interval 0 < x < 180°:
