Advertisements
Advertisements
प्रश्न
Show that `cot(7 1^circ/2) = sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`
Advertisements
उत्तर
We have to prove that `cot(7 1^circ/2) = sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`
L.H.S = `cot(7 1^circ/2)`
= `(cos(7 1^circ/2))/(sin(7 1^circ/2))`
To find `costheta/sintheta`, multiply numerator and denominator by 2 cos θ
Let θ = `71/2^circ`
2θ = 15°
`(2cos^2theta)/(2sin theta cos theta) = (1 + cos 2theta)/(sin 2theta)`
= `(1 + cos 15^circ)/(sin 15^circ)`
= `((1 + sqrt(3) + 1)/(2sqrt(2)))/((sqrt(3) - 1)/(2sqrt(2))`
= `(2sqrt(2) + sqrt(3) + 1)/(sqrt(3) - 1)`
Multiply numerator and denominator by `sqrt(3) + 1`
= `((2sqrt(2) + sqrt(3) + 1)(sqrt(3) + 1))/((sqrt(3) - 1)(sqrt(3) + 1))`
= `(2sqrt(2) + 3 + sqrt(3) + 1)/(3 - 1)`
= `(2sqrt(3) + 2sqrt(2) + 4)/2`
= `(2(sqrt(2) + sqrt(3) + sqrt(6) + 2))/2`
= `sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`
= R.H.S
APPEARS IN
संबंधित प्रश्न
Find all the angles between 0° and 360° which satisfy the equation sin2θ = `3/4`
If sin A = `3/5` and cos B = `9/41, 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of cos(A – B)
Find cos(x − y), given that cos x = `- 4/5` with `pi < x < (3pi)/2` and sin y = `- 24/25` with `pi < y < (3pi)/2`
Prove that cos(π + θ) = − cos θ
Prove that sin(45° + θ) – sin(45° – θ) = `sqrt(2) sin θ`
Show that tan 75° + cot 75° = 4
Prove that cos(A + B) cos C – cos(B + C) cos A = sin B sin(C – A)
Prove that cos 8θ cos 2θ = cos25θ – sin23θ
If cos(α – β) + cos(β – γ) + cos(γ – α) = `- 3/2`, then prove that cos α + cos β + cos γ = sin α + sin β + sin γ = 0
Show that tan(45° − A) = `(1 - tan "A")/(1 + tan "A")`
If tan x = `"n"/("n" + 1)` and tan y = `1/(2"n" + 1)`, find tan(x + y)
If θ + Φ = α and tan θ = k tan Φ, then prove that sin(θ – Φ) = `("k" - 1)/("k" + 1)` sin α
If θ is an acute angle, then find `sin (pi/4 - theta/2)`, when sin θ = `1/25`
Prove that sin 4α = `4 tan alpha (1 - tan^2alpha)/(1 + tan^2 alpha)^2`
Express the following as a sum or difference
cos 5θ cos 2θ
Express the following as a product
sin 75° sin 35°
Show that `((cos theta -cos 3theta)(sin 8theta + sin 2theta))/((sin 5theta - sin theta) (cos 4theta - cos 6theta))` = 1
Prove that `(sin 4x + sin 2x)/(cos 4x + cos 2x)` = tan 3x
Prove that 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x
