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If tan x + \[\tan \left( x + \frac{\pi}{3} \right) + \tan \left( x + \frac{2\pi}{3} \right) = 3\], then prove that \[\frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x} = 1\].
Concept: undefined >> undefined
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
Concept: undefined >> undefined
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If sin (α + β) = 1 and sin (α − β) \[= \frac{1}{2}\], where 0 ≤ α, \[\beta \leq \frac{\pi}{2}\], then find the values of tan (α + 2β) and tan (2α + β).
Concept: undefined >> undefined
If α, β are two different values of x lying between 0 and 2π, which satisfy the equation 6 cos x + 8 sin x = 9, find the value of sin (α + β).
Concept: undefined >> undefined
If \[\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}\] (x ≠ 0), then show that a, b, c and d are in G.P.
Concept: undefined >> undefined
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Concept: undefined >> undefined
If sin α + sin β = a and cos α + cos β = b, show that
Concept: undefined >> undefined
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
Concept: undefined >> undefined
Find three numbers in G.P. whose sum is 38 and their product is 1728.
Concept: undefined >> undefined
If sin α + sin β = a and cos α + cos β = b, show that
Concept: undefined >> undefined
Prove that:
\[\frac{1}{\sin \left( x - a \right) \sin \left( x - b \right)} = \frac{\cot \left( x - a \right) - \cot \left( x - b \right)}{\sin \left( a - b \right)}\]
Concept: undefined >> undefined
Prove that:
Concept: undefined >> undefined
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
Concept: undefined >> undefined
Prove that:
Concept: undefined >> undefined
If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
Concept: undefined >> undefined
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
Concept: undefined >> undefined
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
Concept: undefined >> undefined
If angle \[\theta\] is divided into two parts such that the tangents of one part is \[\lambda\] times the tangent of other, and \[\phi\] is their difference, then show that\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
Concept: undefined >> undefined
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
Concept: undefined >> undefined
If \[\tan\theta = \frac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}\] , then show that \[\sin\alpha + \cos\alpha = \sqrt{2}\cos\theta\].
Concept: undefined >> undefined
