Advertisements
Advertisements
प्रश्न
Prove the following:
tanA + tan(60° + A) + tan(120° + A) = 3 tan 3A
Advertisements
उत्तर
L.H.S. = tanA + tan(60° + A) + tan(120° + A)
= `tan"A" + (tan60^circ + tan"A")/(1 - tan60^circ*tan"A") + (tan120^circ + tan"A")/(1 - tan120^circ*tan"A")`
= `tan"A" + (sqrt(3) + tan"A")/(1 - sqrt(3)tan"A") + (-sqrt(3) + tan"A")/(1 + sqrt(3)tan"A") ...[because tan60^circ = sqrt(3) and tan120^circ = tan(180^circ - 60^circ) = -tan60^circ = -sqrt(3)]`
= `(tan"A"(1 - 3tan^2"A") + (sqrt(3) + tan"A")(1 + sqrt(3)tan"A") + (-sqrt(3) + tan"A")(1 - sqrt(3)tan"A"))/((1 - sqrt(3)tan"A")(1 + sqrt(3)tan"A")`
= `(tan"A" - 3tan^3"A" + sqrt(3) + 3tan"A" + tan"A" + sqrt(3)tan^2"A" - sqrt(3) + 3tan"A" + tan"A" + sqrt(3)tan^2"A")/(1 - 3tan^2"A")`
= `(9tan"A" - 3tan^3"A")/(1 - 3tan^2"A")`
= `3((3tan"A" - tan^3"A")/(1 - 3tan^2"A"))`
= 3 tan 3A
= R.H.S.
APPEARS IN
संबंधित प्रश्न
Prove the following:
`(cos(x - y))/(cos(x + y)) = (cotx coty + 1)/(cotx coty - 1)`
Prove the following:
`(cos15^circ - sin15^circ)/(cos15^circ + sin15^circ) = 1/sqrt(3)`
If sin A = `(-5)/13, pi < "A" < (3pi)/2` and cos B = `3/5, (3pi)/2 < "B" < 2pi` find sin (A + B)
Select the correct option from the given alternatives :
If tan A – tan B = x and cot B – cot A = y, then cot (A – B) = _____
Select the correct option from the given alternatives:
The value of `costheta/(1 + sin theta)` is equal to .....
Prove the following:
If sin 2A = λsin 2B then prove that `(tan("A" + "B"))/(tan("A" - "B")) = (lambda + 1)/(lambda - 1)`
Prove the following:
`(2cos2"A" + 1)/(2cos2"A" - 1)` = tan(60° + A) tan(60° − A)
Prove the following:
3tan610° – 27 tan410° + 33tan210° = 1
The value of sin 163° cos 347° + sin 167° sin 73° is ______
`sqrt3 sin15^circ + cos15^circ` = ______
If A, B, C are the angles of ΔABC, then `tan A/2 tan B/2 + tan B/2 tan C/2 + tan C/2 tan A/2` = ______
If ABCD is a cyclic quadrilateral, then cos A + cos B + cos C + cos D = ______.
If `(cos(x - y))/(cos(x + y)) = ("a" + "b")/("a" - "b")`, then cot x × cot y is equal to ______.
If `2sin(θ + π/3) = cos(θ - π/6)`, then tan θ, = ______.
If sin A + cos A = `sqrt(2)`, then the value of cos2 A is ______.
If A + B = 45°, then (cot A – 1) (cot B – 1) is equal to ______.
If α + β = `π/2` and β + γ = α, then the value of tan α is ______.
`(tan 80^circ - tan 10^circ)/(tan 70^circ)` is equal to ______.
If cos(81° + θ) = `sin(k/3 - θ)`, then k is equal to ______.
lf sin θ = cos θ, then the value of 2 tan2 θ + sin2 θ – 1 is equal to ______.
If cos(A – B) = `3/5` and tan A tan B = 2, then ______.
tan 100° + tan 125° + tan 100° tan 125° = ______.
If tan A – tan B = x and cot B – cot A = y, then cot(A – B) = ______.
cos2 76° + cos2 16° – cos 76° cos 16° is equal to ______.
The value of `sin π/16 sin (3π)/16 sin (5π)/16 sin (7π)/16` is ______.
The value of cot 70° + 4 cos 70° is ______.
If A, B, C, D are the angles of a cyclic quadrilateral, then cos A + cos B + cos C + cos D is equal to ______.
If ABCD is a cyclic quadrilateral, then the value of cos A – cos B + cos C – cos D is equal to ______.
`1/3(sqrt(3) cos 23^circ - sin 23^circ)` is equal to ______.
cos2 x + cos2 y – 2 cos x cos y cos (x + y) is equal to ______.
