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 the values of cot(660°)
Find the value of the trigonometric functions for the following:
cos θ = `- 2/3`, θ lies in the IV quadrant
Find all the angles between 0° and 360° which satisfy the equation sin2θ = `3/4`
Find the value of tan `(7pi)/12`
Show that tan 75° + cot 75° = 4
Prove that cos 8θ cos 2θ = cos25θ – sin23θ
Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B)
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")`
Prove that `tan(pi/4 + theta) tan((3pi)/4 + theta)` = – 1
If θ is an acute angle, then find `sin (pi/4 - theta/2)`, when sin θ = `1/25`
Express the following as a sum or difference
sin 5θ sin 4θ
Show that `((cos theta -cos 3theta)(sin 8theta + sin 2theta))/((sin 5theta - sin theta) (cos 4theta - cos 6theta))` = 1
Prove that sin x + sin 2x + sin 3x = sin 2x (1 + 2 cos x)
Prove that `(sin x + sin 3x + sin 5x + sin 7x)/(cos x + cos x + cos 5x cos 7x)` = tan 4x
If A + B + C = 180°, prove that cos A + cos B − cos C = `- 1 + 4cos "A"/2 cos "B"/2 sin "C"/2`
If A + B + C = 180°, prove that sin2A + sin2B + sin2C = 2 + 2 cos A cos B cos C
Choose the correct alternative:
Let fk(x) = `1/"k" [sin^"k" x + cos^"k" x]` where x ∈ R and k ≥ 1. Then f4(x) − f6(x) =
