Advertisements
Advertisements
Question
Prove the following:
cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
Advertisements
Solution
cot 3x = cot(2x + x) = `(cot 2x cot x-1)/(cot x + cot 2x)` `[∵ cot (A + B) = (cot A cot B - 1)/(cot A + cot B)]`
⇒ cot 3x (cot x + cot 2x) = cot 2x cot x - 1
⇒ cot 3x cot x + cot 3x cot 2x = cot 2x cot x - 1
⇒ cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1
APPEARS IN
RELATED QUESTIONS
Prove the following:
`(cos (pi + x) cos (-x))/(sin(pi - x) cos (pi/2 + x)) = cot^2 x`
Prove the following:
sin2 6x – sin2 4x = sin 2x sin 10x
Prove the following:
`(sin x + sin 3x)/(cos x + cos 3x) = tan 2x`
Prove that: `(cos x + cos y)^2 + (sin x - sin y )^2 = 4 cos^2 (x + y)/2`
If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
If \[\sin A = \frac{12}{13}\text{ and } \sin B = \frac{4}{5}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
sin (A + B)
If \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
sin (A + B)
If \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], where π < A < \[\frac{3\pi}{2}\] and 0 < B <\[\frac{\pi}{2}\], find tan (A + B).
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A + B)
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A - B)
Evaluate the following:
cos 80° cos 20° + sin 80° sin 20°
If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
cos (A + B)
Prove that:
\[\frac{7\pi}{12} + \cos\frac{\pi}{12} = \sin\frac{5\pi}{12} - \sin\frac{\pi}{12}\]
Prove that
Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
Prove that: \[\frac{\sin \left( A + B \right) + \sin \left( A - B \right)}{\cos \left( A + B \right) + \cos \left( A - B \right)} = \tan A\]
Prove that:
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
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).
Prove that:
\[\frac{1}{\sin \left( x - a \right) \sin \left( x - b \right)} = \frac{\cot \left( x - a \right) - \cot \left( x - b \right)}{\sin \left( a - b \right)}\]
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
Find the maximum and minimum values of each of the following trigonometrical expression:
12 cos x + 5 sin x + 4
Reduce each of the following expressions to the sine and cosine of a single expression:
cos x − sin x
Prove that \[\left( 2\sqrt{3} + 3 \right) \sin x + 2\sqrt{3} \cos x\] lies between \[- \left( 2\sqrt{3} + \sqrt{15} \right) \text{ and } \left( 2\sqrt{3} + \sqrt{15} \right)\]
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
If tan \[\alpha = \frac{1}{1 + 2^{- x}}\] and \[\tan \beta = \frac{1}{1 + 2^{x + 1}}\] then write the value of α + β lying in the interval \[\left( 0, \frac{\pi}{2} \right)\]
If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
Express the following as the sum or difference of sines and cosines:
2 sin 4x sin 3x
If cotθ + tanθ = 2cosecθ, then find the general value of θ.
If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a2 - 2b2
[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
State whether the statement is True or False? Also give justification.
If tanA = `(1 - cos B)/sinB`, then tan2A = tanB
