Advertisements
Advertisements
प्रश्न
Prove the following:
`tan^3x/(1 + tan^2x) + cot^3x/(1 + cot^2x)` = secx cosecx − 2sinx cosx
Advertisements
उत्तर
L.H.S. = `tan^3x/(1 + tan^2x) + cot^3x/(1 + cot^2x)`
= `tan^3x/sec^2x + cot^3x/("cosec"^2x)`
= `sin^3x/cos^3x xx cos^2x + cos^3x/sin^3x xx sin^2x`
= `sin^3x/cosx + cos^3x/sinx`
= `(sin^4x + cos^4x)/(sinx*cosx)`
= `((sin^2x + cos^2x)^2 - 2sin^2x cos^2x)/(sinx* cosx)` ...[∵ a2 + b2 = (a + b)2 – 2ab]
= `(1 - 2sin^2x cos^2x)/(sinx*cosx)`
= `1/(sinx* cosx) - (2sin^2xcos^2x)/(sinx*cosx)`
= secx · cosecx – 2sinx · cosx
= R.H.S.
APPEARS IN
संबंधित प्रश्न
Prove the following:
sin [(n + 1)A]. sin [(n + 2)A] + cos [(n + 1)A]. cos [(n + 2)A] = cos A
Prove the following:
`(cos(x - y))/(cos(x + y)) = (cotx coty + 1)/(cotx coty - 1)`
If tan A = `5/6, tan "B" = 1/11`, prove that A + B = `pi/4`
Select the correct option from the given alternatives :
The value of sin(n + 1) A sin (n + 2) A + cos(n + 1) A cos(n + 2) A is equal to
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:
tanA + tan(60° + A) + tan(120° + A) = 3 tan 3A
Prove the following:
3tan610° – 27 tan410° + 33tan210° = 1
Prove the following:
tan A + 2 tan 2A + 4 tan 4A + 8 cot 8A = cot A
tan A +2 tan 2A + 4 tan 4A + 8 cot 8A = ?
If f(x) = log (sec x + tan x), then `"f'"(π/4)` = ____________.
In Δ ABC, if tan A + tan B + tan C = 6 and tan A tan B = 2 then tan C = ______.
`sqrt3 sin15^circ + cos15^circ` = ______
The imaginary part of `1/(1 - sintheta + icostheta)` is equal to ______
`(sin8A + sin2A)/(cos2A - cos8A)` 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 `0 < β < α < π/4, cos (α + β) = 3/5` and cos (α – β) = `4/5`, then sin 2α is equal to ______.
If cos (α + β) = `4/5` and sin (α – β) = `5/13`, where `0 ≤ α, β ≤ π/4`, then tan 2α is equal to ______.
`(tan 80^circ - tan 10^circ)/(tan 70^circ)` is equal to ______.
lf sin θ = cos θ, then the value of 2 tan2 θ + sin2 θ – 1 is equal to ______.
The expression cos2(A – B) + cos2 B – 2 cos(A – B) cos A cos B is ______.
If cos(A – B) = `3/5` and tan A tan B = 2, then ______.
If `π/2 < α < π, π < β < (3π)/2`; sin α = `15/17` and tan β = `12/5`, then the value of sin(β – α) is ______.
sin 4θ can be written as ______.
cos2 76° + cos2 16° – cos 76° cos 16° is equal to ______.
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 ______.
tan 57° – tan 12° – tan 57° tan 12° is equal to ______.
The value of tan 3A – tan 2A – tan A is ______.
