If θ is an acute angle and tan θ + cot θ = 2 , find the value of tan 7 θ + cot 7 θ .
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tan θ + cot θ = 2 ⇒ sin θ cos θ sin 2 θ + cos 2 θ = 2 1 = 2 sin θ cos θ ⇒ 1 = sin 2 θ ⇒ θ = 4 π ⇒ tan 7 4 π + cot 7 4 π = 2
Small note: θ = 4 π + k π where k is an integer. However, since tan and cot have period π , this doesn't raise any issues.
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Can you explain in a bit more?
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Sure, the tan and cot graphs have period π , which means that tan ( θ ) ≡ tan ( θ + k π ) where k is an integer. In other words, the tan graph repeats itself after every 1 8 0 ∘ .
Similarly, the sin graph repeats itself every 2 π , so 1 = sin 2 θ ⇒ 2 θ = 2 π + 2 k π ⇒ θ = 4 π + k π as I mentioned.
I always find these types of questions easier when visualising a sketch. Think about the line y = 1 and y = sin x . An obvious solution is 2 π , but there's another solution every ± 2 k π .
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Relevant wiki: Trigonometric Equations - Problem Solving - Easy
tan θ + tan θ 1 = 2 ( ∵ cot θ = tan θ 1 )
⟹ ( tan θ − 1 ) 2 = 0 ⟹ tan θ = 1
tan 7 θ + cot 7 θ = 1 + 1 = 2