Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app

JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

JEE Main · Mathematics · Trigonometric Ratios & Identities

If (x-y), x, (x+y) are three distinct numbers which are in harmonic...

If

\cos (x-y), \cos x, \cos (x+y)

are three distinct numbers which are in harmonic progression and

\cos x \neq \cos y

, then

1+\cos y

is equal to

Options

A $\cos ^2 x$
B $-\cos ^2 x$
C $\cos ^2 x-1$
D $\cos ^2 x-2$

Correct answer

A. $\cos ^2 x$

Step-by-step solution

\cos (x-y), \cos x, \cos (x+y)

are in HP. Then,

\cos x=\frac{2 \cos (x-y) \cos (x+y)}{\cos (x+y)+\cos (x-y)}\cos x=\frac{\cos 2 x+\cos 2 y}{2 \cos x \cdot \cos y}\cos x=\frac{2 \cos ^2 x+2 \cos ^2 y-2}{2 \cos x \cdot \cos y}\cos ^2 x \cdot \cos y=\cos ^2 x+\cos ^2 y-1\cos ^2 x(\cos y-1)=\left(\cos ^2 y-1\right)\cos ^2 x(1-\cos y)=\left(1-\cos ^2 y\right)\cos ^2 x\left(2 \sin ^2 \frac{y}{2}\right)=\sin ^2 y\cos ^2 x\left(2 \sin ^2 \frac{y}{2}\right)=\left(2 \sin \frac{y}{2} \cdot \cos \frac{y}{2}\right)^2\cos ^2 x\left(2 \sin ^2 \frac{y}{2}\right)=4 \sin ^2 \frac{y}{2} \cdot \cos ^2 \frac{y}{2}\cos ^2 x=\left(2 \cos ^2 \frac{y}{2}-1\right)+1\cos y+1=\cos ^2 x

or

\quad 1+\cos y=\cos ^2 x

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 Highly selective Backlog Qs for JEE Main PYQs