Advertisements
Advertisements
Question
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
Advertisements
Solution
\[\text{ We know that }36^\circ + 9^\circ = 45^\circ\]
Therefore,
\[ \tan\left( 36^\circ + 9^\circ \right) = \tan45^\circ\]
\[ \Rightarrow \frac{\tan36^\circ + \tan9^\circ}{1 - \tan36^\circ \tan9^\circ} = 1\]
\[ \Rightarrow \tan36^\circ + \tan9^\circ = 1 - \tan36^\circ \tan9^\circ\]
\[ \Rightarrow \tan36^\circ + \tan9^\circ + \tan36^\circ \tan9^\circ = 1\]
Hence proved.
APPEARS IN
RELATED QUESTIONS
Prove that `cot^2 pi/6 + cosec (5pi)/6 + 3 tan^2 pi/6 = 6`
Prove the following:
cos2 2x – cos2 6x = sin 4x sin 8x
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove that: `(cos x + cos y)^2 + (sin x - sin y )^2 = 4 cos^2 (x + y)/2`
Prove that: sin 3x + sin 2x – sin x = 4sin x `cos x/2 cos (3x)/2`
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:
cos (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 \[\sin A = \frac{1}{2}, \cos B = \frac{12}{13}\], where \[\frac{\pi}{2}\]< A < π and \[\frac{3\pi}{2}\] < B < 2π, find tan (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:
Prove that:
If \[\tan A = \frac{m}{m - 1}\text{ and }\tan B = \frac{1}{2m - 1}\], then prove that \[A - B = \frac{\pi}{4}\].
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:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
If tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) \[\frac{1}{a} - \frac{1}{b}\].
If sin α + sin β = a and cos α + cos β = b, show that
If sin α + sin β = a and cos α + cos β = b, show that
If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
Write the interval in which the value of 5 cos x + 3 cos \[\left( x + \frac{\pi}{3} \right) + 3\] lies.
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
The value of \[\sin^2 \frac{5\pi}{12} - \sin^2 \frac{\pi}{12}\]
If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
tan 3A − tan 2A − tan A =
If tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
If tan (A − B) = 1 and sec (A + B) = \[\frac{2}{\sqrt{3}}\], the smallest positive value of B is
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
If α and β are the solutions of the equation a tan θ + b sec θ = c, then show that tan (α + β) = `(2ac)/(a^2 - c^2)`.
Find the most general value of θ satisfying the equation tan θ = –1 and cos θ = `1/sqrt(2)`.
If cos(θ + Φ) = m cos(θ – Φ), then prove that 1 tan θ = `(1 - m)/(1 + m) cot phi`
[Hint: Express `(cos(theta + Φ))/(cos(theta - Φ)) = m/1` and apply Componendo and Dividendo]
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.
Given x > 0, the values of f(x) = `-3cos sqrt(3 + x + x^2)` lie in the interval ______.
State whether the statement is True or False? Also give justification.
If tanA = `(1 - cos B)/sinB`, then tan2A = tanB
State whether the statement is True or False? Also give justification.
If tanθ + tan2θ + `sqrt(3)` tanθ tan2θ = `sqrt(3)`, then θ = `("n"pi)/3 + pi/9`
