Advertisements
Advertisements
Question
If A + B + C = 180°, prove that `tan "A"/2 tan "B"/2 + tan "B"/2 tan "C"/2 + tan "C"/2 tan "A"/2` = 1
Advertisements
Solution
Given A + B + C = 180°
⇒ `("A" + "" + "C")/2` = 90°
So `tan(("A" + "B")/2) = tan(90^circ - "C"/2) = cot "C"/2`
(i.e) `(tan "A"/2 + tan "B"/2)/(1 - tan "A"/2 tan "B"/2) = cot "C"/2 = 1/(tan "C"/2)`
⇒ `(tan "A"/2 + tan "B"/2)tan "C"/2 = 1 - tan "A"/2 tan "B"/2`
(i.e) `tan "A"/2 tan "C"/2 + tan "B"/2 tan "C"/2 = 1 - tan "A"/2 tan "B"/2`
(i.e) `tan "A"/2 tan "B"/2 + tan "B"/2 tan "C"/2 + tan "C"/2 tan "A"/2` = 1
APPEARS IN
RELATED QUESTIONS
Find the values of sin (– 1110°)
Find the values of `sin (-(11pi)/3)`
`(5/7, (2sqrt(6))/7)` is a point on the terminal side of an angle θ in standard position. Determine the six trigonometric function values of angle θ
Find all the angles between 0° and 360° which satisfy the equation sin2θ = `3/4`
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2` find the value of sin(x + y)
Find sin(x – y), given that sin x = `8/17` with 0 < x < `pi/2`, and cos y = `- 24/25`, x < y < `(3pi)/2`
Prove that cos(π + θ) = − cos θ
Prove that sin(30° + θ) + cos(60° + θ) = cos θ
If x cos θ = `y cos (theta + (2pi)/3) = z cos (theta + (4pi)/3)`. find the value of xy + yz + zx
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
Prove that cos 5θ = 16 cos5θ – 20 cos3θ + 5 cos θ
Express the following as a sum or difference
sin 4x cos 2x
Show that sin 12° sin 48° sin 54° = `1/8`
Show that `((cos theta -cos 3theta)(sin 8theta + sin 2theta))/((sin 5theta - sin theta) (cos 4theta - cos 6theta))` = 1
If A + B + C = `pi/2`, prove the following sin 2A + sin 2B + sin 2C = 4 cos A cos B cos C
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that sin2 B + sin2 C = 1
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that cos B – cos C = `- 1 + 2sqrt(2) cos "B"/2 sin "C"/2`
Choose the correct alternative:
`(sin("A" - "B"))/(cos"A" cos"B") + (sin("B" - "C"))/(cos"B" cos"C") + (sin("C" - "A"))/(cos"C" cos"A")` is
