Advertisements
Advertisements
प्रश्न
In ΔABC, A + B + C = π show that
cos A + cos B – cos C = `4cos "A"/2 cos "B"/2 sin "C"/2 - 1`
Advertisements
उत्तर
L.H.S. = cos A + cos B – cos C
= `2cos(("A" + "B")/2) cos(("A" - "B")/2) - [1 - 2sin^2("C"/2)]`
= `2cos(pi/2 - "C"/2)cos(("A" - "B")/2) - 1 + 2sin^2("C"/2)` ...[∵ A + B + C = π]
= `2sin("C"/2)cos(("A" - "B")/2) + 2sin^2("C"/2) - 1`
= `2sin("C"/2)[cos(("A" - "B")/2) + sin^2("C"/2)] - 1`
= `2sin("C"/2)[cos(("A" - "B")/2) + sin{pi/2 - (("A" + "B")/2)}] - 1` ...[∵ A + B + C = π]
= `2sin("C"/2)[cos(("A" - "B")/2) + cos(("A" + "B")/2)] - 1`
= `2sin("C"/2) xx 2cos[(("A" + "B")/2 + ("A" - "B")/2)/2]*cos[(("A" + "B")/2 - ("A" - "B")/2)/2] - 1`
= `2sin("C"/2) xx 2cos("A"/2)*cos("B"/2) - 1`
= `4cos("A"/2)*cos("B"/2)*sin("C"/2) - 1`
= R.H.S.
APPEARS IN
संबंधित प्रश्न
In ΔABC, A + B + C = π show that
cos 2A + cos 2B + cos 2C = –1 – 4 cos A cos B cos C
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
`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:
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`
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` = ______
`(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 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 ______.
In a ΔABC, `cos((B + 2C + 3A)/2) + cos((A - B)/2)` is ______.
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 ______.
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 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 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 ______.
If A + B + C = 180°, then `(sin 2A + sin 2B + sin 2C)/(cos A + cos B + cos C - 1)` is equal to ______.
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 ______.
