Advertisements
Advertisements
Question
In ΔABC, A + B + C = π show that
tan 2A + tan 2B + tan 2C = tan 2A tan 2B tan 2C
Advertisements
Solution
In ΔABC,
A + B + C = π
∴ 2A + 2B + 2C = 2π
∴ 2A + 2B = 2π – 2C
∴ tan (2A + 2B) = tan (2π – 2C)
∴ `(tan2"A" + tan2"B")/(1 - tan2"A"*tan2"B")` = – tan 2C
∴ tan 2A + tan 2B = − tan 2C . (1 − tan 2A . tan 2B)
∴ tan 2A + tan 2B = – tan 2C + tan 2A· tan 2B· tan 2C
∴ tan 2A + tan 2B + tan 2C = tan 2A · tan 2B· tan 2C
APPEARS IN
RELATED QUESTIONS
In ΔABC, A + B + C = π show that
cos A + cos B – cos C = `4cos "A"/2 cos "B"/2 sin "C"/2 - 1`
In ΔABC, A + B + C = π show that
`sin^2 "A"/2 + sin^2 "B"/2 - sin^2 "C"/2 = 1 - 2cos "A"/2 cos "B"/2 sin "C"/2`
In ΔABC, A + B + C = π show that
cos2A +cos2B – cos2C = 1 – 2 sin A sin B cos C
Prove the following:
If sin α sin β − cos α cos β + 1 = 0 then prove cot α tan β = −1
Prove the following:
`cos (2pi)/15 cos (4pi)/15cos (8pi)/15cos (16pi)/15 = 1/16`
Prove the following:
`(1 + cos pi/8)(1 + cos (3pi)/8)(1 + cos (5pi)/8)(1 + cos (7pi)/8) = 1/8`
Prove the following:
In any triangle ABC, sin A − cos B = cos C then ∠B = `pi/2`.
Prove the following:
In ∆ABC, ∠C = `(2pi)/3`, then prove that cos2A + cos2B − cos A cos B = `3/4`
If A and Bare supplementary angles, then `sin^2 "A"/2 + sin^2 "B"/2` = ______.
In a ΔABC, A : B : C = 3 : 5 : 4. Then `a + b + csqrt2` is equal to ______
The value of `[(1 - cos pi/6 + isin pi/6)/(1 - cos pi/6 - isin pi/6)]^6` = ______
If `cos "A" = 3/4,`then 32 sin`"A"/2 cos (5"A")/2` = ?
`(sin20^circ +2sin40^circ)/sin70^circ=` ______.
If A, B, C are the angles of ΔABC then cotA.cotB + cotB. cotC + cotC + cotA = ______.
If A + B + C = π, then sin 2A + sin 2B + sin 2C is equal to ______.
If α + β – γ = π, then sin2 α + sin2 β – sin2 γ 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 ______.
ΔABC is a right angled isosceles triangle with ∠B = 90°. If D is a point on AB, ∠CDB = 15° and AD = 35 cm, then CD is equal to ______.
If A + B + C = π and sin C + sin A cos B = 0, then tan A . cot B is equal to ______.
If A + B + C = π(A, B, C > 0) and the ∠C is obtuse, then ______.
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 are the angles of a triangle, then sin2 A + sin2 B + sin2 C – 2 cos A cos B cos C is equal to ______.
If A + B + C = 180°, then `(sin 2A + sin 2B + sin 2C)/(cos A + cos B + cos C - 1)` is equal to ______.
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 ______.
The value of `tan A/2 tan B/2 + tan B/2 tan C/2 + tan C/2 tan A/2` is ______.
