Advertisements
Advertisements
प्रश्न
If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
Advertisements
उत्तर
Given:
\[\sin\alpha \sin\beta - \cos \alpha \cos \beta + 1 = 0 \]
\[ \Rightarrow - (\cos\alpha \cos\beta - \sin\alpha \sin\beta) + 1 = 0\]
\[ \Rightarrow - \cos(\alpha + \beta) + 1 = 0\]
\[ \Rightarrow \cos(\alpha + \beta) = 1\]
\[\text{ Therefore, }\sin(\alpha + \beta) = 0 . . . . (1) (\text{ Since }\sin\theta = \sqrt{1 - \cos^2 \theta} ) \]
Hence ,
\[1 + \cot\alpha \tan\beta = 1 + \frac{\cos\alpha \sin\beta}{\sin\alpha \cos\beta} \]
\[ = \frac{\sin\alpha\cos\beta + \cos\alpha\sin\beta}{\sin\alpha \cos\beta}\]
\[ = \frac{\sin(\alpha + \beta)}{\sin\alpha\cos\beta}\]
\[ = 0 . . . \left\{\text{ From eq }(1) \right\}\]
Hence proved .
APPEARS IN
संबंधित प्रश्न
Prove that `cot^2 pi/6 + cosec (5pi)/6 + 3 tan^2 pi/6 = 6`
Prove the following: `cos (pi/4 xx x) cos (pi/4 - y) - sin (pi/4 - x)sin (pi/4 - y) = sin (x + y)`
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove that: `(cos x - cosy)^2 + (sin x - sin y)^2 = 4 sin^2 (x - y)/2`
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
Prove that: `((sin 7x + sin 5x) + (sin 9x + sin 3x))/((cos 7x + cos 5x) + (cos 9x + cos 3x)) = tan 6x`
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:
cos (A + B)
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:
sin (A − B)
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 \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], 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)
Evaluate the following:
cos 47° cos 13° − sin 47° sin 13°
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:
Prove that:
\[\frac{\tan^2 2x - \tan^2 x}{1 - \tan^2 2x \tan^2 x} = \tan 3x \tan x\]
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
If sin α + sin β = a and cos α + cos β = b, show that
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
If x cos θ = y cos \[\left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)\]then write the value of \[\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\]
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
Write the interval in which the value of 5 cos x + 3 cos \[\left( x + \frac{\pi}{3} \right) + 3\] lies.
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 =
If cot (α + β) = 0, sin (α + 2β) is equal to
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
If cos (A − B) \[= \frac{3}{5}\] and tan A tan B = 2, then
If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
If f(x) = cos2x + sec2x, then ______.
[Hint: A.M ≥ G.M.]
The value of tan3A - tan2A - tanA is equal to ______.
The value of sin(45° + θ) - cos(45° - θ) is ______.
The maximum distance of a point on the graph of the function y = `sqrt(3)` sinx + cosx from x-axis is ______.
