Advertisements
Advertisements
Question
Prove the following:
If sin α sin β − cos α cos β + 1 = 0 then prove cot α tan β = −1
Advertisements
Solution
sin α sin β − cos α cos β + 1 = 0
∴ cos α cos β − sin α sin β = 1
∴ cos (α + β) = 1
∴ α + β = 0 ...[∵ cos 0 = 1]
∴ β = − α
L.H.S. = cot α tan β
= cot α tan (− α)
= − cot α tan α
= − 1
= R.H.S.
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
`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
`tan "A"/2 tan "B"/2 + tan "B"/2 tan "C"/2 + tan "C"/2tan "A"/2` = 1
In ΔABC, A + B + C = π show that
tan 2A + tan 2B + tan 2C = tan 2A tan 2B tan 2C
In ΔABC, A + B + C = π show that
cos2A +cos2B – cos2C = 1 – 2 sin A sin B cos C
Select the correct option from the given alternatives :
In ∆ABC if cot A cot B cot C > 0 then the triangle is _________
Prove the following:
`(1 + cos pi/8)(1 + cos (3pi)/8)(1 + cos (5pi)/8)(1 + cos (7pi)/8) = 1/8`
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 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`
The value of `[(1 - cos pi/6 + isin pi/6)/(1 - cos pi/6 - isin pi/6)]^6` = ______
If A + B = C, then cos2 A + cos2 B + cos2 C – 2 cos A cos B cos C is equal to ______.
If α + β – γ = π, then sin2 α + sin2 β – sin2 γ 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 ______.
If A + B + C = π, then cos2 A + cos2 B + cos2 C is equal to ______.
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 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 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 are the angles of a triangle, then sin2 A + sin2 B + sin2 C – 2 cos A cos B cos C is equal to ______.
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 ______.
lf A + B + C = π, then `cosA/(sinBsinC) + cosB/(sinCsinA) + cosC/(sinAsinB)` is equal to ______.
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 ______.
