For 0 < θ < π /2 , which options solve ∑ m = 1 6 csc ( θ + 4 ( m − 1 ) π ) csc ( θ + 4 mπ ) = 4 2 ?

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MathematicsTrigonometric Ratios & IdentitiesAddition, Multiplication, Subtraction formula based,Multiple/Submultiple anglesMedium2 minQB

MathematicsMediummultiple choice

For $0 < θ < π /2$ , which options solve $\sum_{m = 1}^{6} csc (θ + \frac{( m - 1 ) π}{4}) csc (θ + \frac{mπ}{4}) = 42$ ?

Options:(select one or more)

Answer:

C, D

Solution:

$csc a csc (a + π /4) = 2 [cot a - cot (a + π /4)]$ . The sum telescopes to $2 [cot θ - cot (θ + 3 π /2)] = 2 (cot θ + tan θ)$ . Hence $sin 2 θ = 1/2$ , giving $θ = π /12$ or $5 π /12$ .

Stream:JEESubject:MathematicsTopic:Trigonometric Ratios & IdentitiesSubtopic:Addition, Multiplication, Subtraction formula based,Multiple/Submultiple angles

⏱ 2mℹ️ Source: QB

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