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(480°)
Find the values of cot(660°)
Find the value of the trigonometric functions for the following:
sec θ = `13/5`, θ lies in the IV quadrant
Show that `sin^2 pi/18 + sin^2 pi/9 + sin^2 (7pi)/18 + sin^2 (4pi)/9` = 2
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)
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of cos(x − y)
Find the value of cos 105°.
Prove that sin(n + 1) θ sin(n – 1) θ + cos(n + 1) θ cos(n – 1)θ = cos 2θ, n ∈ Z
Prove that cos 8θ cos 2θ = cos25θ – sin23θ
If cos θ = `1/2 ("a" + 1/"a")`, show that cos 3θ = `1/2 ("a"^3 + 1/"a"^3)`
If A + B = 45°, show that (1 + tan A)(1 + tan B) = 2
Express the following as a sum or difference
2 sin 10θ cos 2θ
Express the following as a sum or difference
cos 5θ cos 2θ
Prove that sin x + sin 2x + sin 3x = sin 2x (1 + 2 cos x)
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
If A + B + C = `pi/2`, prove the following cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B sin C
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that sin2 B + sin2 C = 1
Choose the correct alternative:
`1/(cos 80^circ) - sqrt(3)/(sin 80^circ)` =
