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Prove that:
\[\frac{\tan^2 2x - \tan^2 x}{1 - \tan^2 2x \tan^2 x} = \tan 3x \tan x\]
Concept: undefined >> undefined
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
Concept: undefined >> undefined
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Prove that sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
Concept: undefined >> undefined
If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.
Concept: undefined >> undefined
If tan A = x tan B, prove that
\[\frac{\sin \left( A - B \right)}{\sin \left( A + B \right)} = \frac{x - 1}{x + 1}\]
Concept: undefined >> undefined
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
Concept: undefined >> undefined
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
Concept: undefined >> undefined
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
Concept: undefined >> undefined
If tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) \[\frac{1}{a} - \frac{1}{b}\].
Concept: undefined >> undefined
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Concept: undefined >> undefined
If x lies in the first quadrant and \[\cos x = \frac{8}{17}\], then prove that:
Concept: undefined >> undefined
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
Concept: undefined >> undefined
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
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
