Advertisements
Advertisements
Question
In ΔABC, A + B + C = π show that
`cot "A"/2 + cot "B"/2 + cot "C"/2 = cot "A"/2 cot "B"/2 cot "C"/2`
Advertisements
Solution
In ΔABC,
A + B + C = π
∴ A + B = π – C
∴ `tan(("A" + "B")/2) = tan((pi - "C")/2)`
∴ `tan("A"/2 + "B"/2) = tan(pi/2 - "C"/2)`
∴ `(tan "A"/2 + tan "B"/2)/(1 - tan "A"/2*tan "B"/2) = cot "C"/2`
∴ `(tan "A"/2 + tan "B"/2)/(1 - tan "A"/2*tan "B"/2) = 1/(tan "C"/2)`
∴ `tan "C"/2*(tan "A"/2 + tan "B"/2) = 1 - tan "A"/2*tan "B"/2`
∴ `tan "B"/2*tan "C"/2 + tan "A"/2*tan "C"/2 + tan "A"/2*tan "B"/2` = 1
Dividing throughout by `tan "A"/2*tan "B"/2*tan "C"/2`, we get
`1/(tan "A"/2) + 1/(tan "B"/2) + 1/(tan "C"/2) = 1/(tan "A"/2*tan "B"/2*tan "C"/2)`
∴ `cot "A"/2 + cot "B"/2 + cot "C"/2 = cot "A"/2 cot "B"/2 cot "C"/2`
APPEARS IN
RELATED QUESTIONS
In ΔABC, A + B + C = π show that
sin A + sin B + sin C = `4cos "A"/2 cos "B"/2 cos "C"/2 `
In ΔABC, A + B + C = π show that
sin2A + sin2B − sin2C = 2 sin A sin B cos C
In ΔABC, A + B + C = π show that
cos2A +cos2B – cos2C = 1 – 2 sin A sin B cos C
Prove the following:
If A + B + C = `(3pi)/2`, then cos 2A + cos 2B + cos 2C = 1 − 4 sin A sin B sin C
Prove the following:
In ∆ABC, ∠C = `(2pi)/3`, then prove that cos2A + cos2B − cos A cos B = `3/4`
The area of the Δ ABC is `10sqrt3` cm2, angle B is 60° and its perimeter is 20 cm , then l(AC) = ______.
In a ΔABC, A : B : C = 3 : 5 : 4. Then `a + b + csqrt2` is equal to ______
If `cos "A" = 3/4,`then 32 sin`"A"/2 cos (5"A")/2` = ?
`(sin20^circ +2sin40^circ)/sin70^circ=` ______.
If A + B + C = π, then sin 2A + sin 2B + sin 2C is equal to ______.
If A + B = C, then cos2 A + cos2 B + cos2 C – 2 cos A cos B cos C is equal to ______.
Let A, B and C are the angles of a triangle and `tan(A/2) = 1/3, tan(B/2) = 2/3`. Then, `tan(C/2)` is equal to ______.
If A + B + C = 270°, then cos 2A + cos 2B + cos 2C is equal to ______.
If A + B + C = π, then sin 2A + sin 2B – sin 2C is equal to ______.
In a ΔABC, if cos A cos B cos C = `(sqrt(3) - 1)/8` and sin A sin B sin C = `(3 + sqrt(3))/8`, then the angles of the triangle are ______.
If A + B + C = π, then cos2 A + cos2 B + cos2 C is equal to ______.
If sin A + sin B = C, cos A + cos B = D, then the value of sin(A + B) = ______.
If A + B + C = π and sin C + sin A cos B = 0, then tan A . cot B is equal to ______.
If x + y + z = 180°, then cos 2x + cos 2y – cos 2z is equal to ______.
If A + B + C = π(A, B, C > 0) and the ∠C is obtuse, then ______.
If a ΔABC, the value of sin A + sin B + sin C is ______.
If A + B + C = 270°, then cos 2A + cos 2B + cos 2C + 4 sin A sin B sin C is equal to ______.
If A + B = C = 180°, then the value of `cot A/2 + cot B/2 + cot C/2` will be ______.
In any ΔABC, if tan A + tan B + tan C = 6 and tan A tan B = 2, then the values of tan A, tan B and tan C are ______.
If cos A = cos B cos C and A + B + C = π, then the value of cot B cot C is ______.
The value of cot A cot B + cot B cot C + cot C cot A is ______.
