Advertisements
Advertisements
Question
Show that `cot(7 1^circ/2) = sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`
Solution
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
RELATED QUESTIONS
Find the values of sin(480°)
Find the values of sin (– 1110°)
Find the values of `sin (-(11pi)/3)`
Find the value of the trigonometric functions for the following:
sec θ = `13/5`, θ lies in the IV quadrant
Find the value of sin 105°
Prove that cos(π + θ) = − cos θ
Prove that sin(π + θ) = − sin θ.
Find a quadratic equation whose roots are sin 15° and cos 15°
Prove that sin(45° + θ) – sin(45° – θ) = `sqrt(2) sin θ`
Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B)
Show that tan(45° + A) = `(1 + tan"A")/(1 - tan"A")`
Find the value of cos 2A, A lies in the first quadrant, when cos A = `15/17`
If A + B = 45°, show that (1 + tan A)(1 + tan B) = 2
Show that `cos pi/15 cos (2pi)/15 cos (3pi)/15 cos (4pi)/15 cos (5pi)/15 cos (6pi)/15 cos (7pi)/15 = 1/128`
Prove that 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x
If A + B + C = `pi/2`, prove the following sin 2A + sin 2B + sin 2C = 4 cos A cos B cos C
Choose the correct alternative:
If `pi < 2theta < (3pi)/2`, then `sqrt(2 + sqrt(2 + 2cos4theta)` equals to
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) =
