The value of the expression 1 + cosec(π)/(4) + cosec(π)/(8) + cosec(π)/(16) is equal to

Correct answer: C. cot(π)/(32) We know that cosecθ + cot θ = (1 + cos θ)/(sin θ) = 2cos^ 2 (θ)/(2) 2sin(θ)/(2)cos(θ)/(2) = cot(θ)/(2) ⇒ cosecθ = cot(θ)/(2) - cot θ ⇒ cosec(π)/(4) = cot(π)/(8) - cot(π)/(4) cosec(π)/(8) = cot(π)/(16) - cot(π)/(8) cosec(π)/(16) = cot(π)/(32) - cot(π)/(16) Adding them, we get, cosec(π)/(4) + cosec(π)/(8) + cosec(π)/(16) = cot(π)/(32) - cot(π)/(4) ⇒ 1 + cosec(π)/(4) + cosec(π)/(8) + cosec(π)/(16) = cot(π)/(32)

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

NTA Abhyas 2020 · Mathematics · Trigonometric Ratios & Identities

The value of the expression 1 + cosec(π)/(4) + cosec(π)/(8) +...

The value of the expression

1 + cosec\dfrac{\pi}{4} + cosec\dfrac{\pi}{8} + cosec\dfrac{\pi}{16}

is equal to

Options

A $cot\dfrac{\pi}{8}$
B $cot\dfrac{\pi}{16}$
C $cot\dfrac{\pi}{32}$
D ${cosec}^{2}\dfrac{\pi}{16}$

Correct answer

C. $cot\dfrac{\pi}{32}$

Step-by-step solution

We know that

cosec\theta + cot \theta = \dfrac{1 + cos \theta}{sin \theta} = \dfrac{2cos^{2}\dfrac{θ}{2}}{2sin\dfrac{θ}{2}cos\dfrac{θ}{2}} = cot\dfrac{θ}{2}

⇒ cosec\theta = cot\dfrac{θ}{2} - cot \theta

⇒ cosec\dfrac{\pi}{4} = cot\dfrac{\pi}{8} - cot\dfrac{\pi}{4}

cosec\dfrac{\pi}{8} = cot\dfrac{\pi}{16} - cot\dfrac{\pi}{8}

cosec\dfrac{\pi}{16} = cot\dfrac{\pi}{32} - cot\dfrac{\pi}{16}

Adding them, we get,

cosec\dfrac{\pi}{4} + cosec\dfrac{\pi}{8} + cosec\dfrac{\pi}{16} = cot\dfrac{\pi}{32} - cot\dfrac{\pi}{4}

⇒ 1 + cosec\dfrac{\pi}{4} + cosec\dfrac{\pi}{8} + cosec\dfrac{\pi}{16} = cot\dfrac{\pi}{32}

Practice Trigonometric Ratios & Identities on Quantrex Academy →

More from Trigonometric Ratios & Identities

Let α = (1 - 2 ((π)/(11)))(1 - 2 ((3π)/(11)))(1 - 2 ((9π)/(11)))(1 - 2 ((27π)/(11)))(1… JEE Advanced 2026 If A = ( 3^∘)/( 9^∘) + ( 9^∘)/( 27^∘) + ( 27^∘)/( 81^∘) and B = 81^∘ - 3^∘ , then (B)/(A)… JEE Main 2026 Let P = \ θ ∈ [0, 4π] : ^2θ ≠ 1\ and S = \ a ∈ ℤ : 2( ^8θ - ^8θ) 2θ = a^2, θ ∈ P\ . Then… JEE Main 2026 If ((π)/(18)) ((5π)/(18)) ((7π)/(18)) = K , then the value of ((10Kπ)/(3)) is : JEE Main 2026 If (A-B) A + ^ 2 C ^ 2 ~A =1, ~A, ~B, C ∈(0, (π)/(2)) , then JEE Main 2026 If x=(5)/(12) for some x ∈(π, (3 π)/(2)) , then 7 x( (13 x)/(2)+ (13 x)/(2))+ 7 x( (13… JEE Main 2026 The value of 3 cosec 20^ ∘ - 20^ ∘ 20^ ∘ 40^ ∘ 60^ ∘ 80^ ∘ is equal to JEE Main 2026 The least value of ( ^ 2 θ-6 θ θ+3 ^ 2 θ+2) is JEE Main 2026 Let (π)/(2) ((15 θ)/(2))( 8 θ+ 8 θ)+ ((15 θ)/(2))( 8 θ- 8 θ) is equal to JEE Main 2026 Let α and β respectively be the maximum and the minimum values of the function f(θ)=4( ^… JEE Main 2026 Let (α+β)=-(1)/(10) and (α-β)=(3)/(8) , where 0<α<(π)/(3) and 0<β<(π)/(4) . If 2 α= 3(1-r… JEE Main 2026 If ^ 2 48^ ∘ - ^ 2 12^ ∘ ^ 2 24^ ∘ - ^ 2 6^ ∘ = α+β 5 2 , where α, β ∈ ℕ , then α+β is… JEE Main 2026 The value of cosec 10^ ∘ - 3 10^ ∘ is equal to : JEE Main 2026 Let α= 1 60^ ∘ 61^ ∘ + 1 62^ ∘ 63^ ∘ +…+ 1 118^ ∘ 119^ ∘ Then the value of ( cosec 1^ ∘ α… JEE Advanced 2025 If x+ ^2 x=1, x ∈(0, (π)/(2)) , then ( ^ 12 x+ ^ 12 x)+3( ^ 10 x+ ^ 10 x+ ^8 x+ ^8 x)+(… JEE Main 2025 If ∑_ r=1 ^ 13 \ (1)/( ((π)/(4)+(r-1) (π)/(6)) ((π)/(4)+(r π)/(6)))\ =a 3 +b, a, b ∈ Z… JEE Main 2025 Full Trigonometric Ratios & Identities list Trigonometric Ratios & Identities JEE Maths PYQs JEE Mathematics PYQs All NTA Abhyas JEE Main PYQs