Advertisements
Advertisements
प्रश्न
Prove that
Advertisements
उत्तर
\[\text{ LHS }= \frac{\cos9^\circ + \sin9^\circ}{\cos9^\circ - \sin9^\circ}\]
\[ = \frac{\frac{\cos9^\circ}{\cos9^\circ} + \frac{\sin9^\circ}{\cos9^\circ}}{\frac{\cos9^\circ}{\cos9^\circ} - \frac{\sin9^\circ}{\cos9^\circ}} \left(\text{ Dividing the numerator and denominator by }\cos9 \right)\]
\[ = \frac{1 + \tan9^\circ}{1 - \tan9^\circ}\]
\[ = \frac{1 + \tan9^\circ}{1 + 1 \times \tan9^\circ}\]
\[ = \frac{\tan45^\circ + \tan9^\circ}{1 - \tan45^\circ \times \tan9^\circ} \left(\text{ As }\tan45^\circ = 1 \right)\]
\[ = \tan\left( 45^\circ + 9^\circ \right) \left[\text{ As }\frac{\tan A + \tan B}{1 - \tan A \tan B} = \tan\left( A + B \right) \right]\]
\[ = \tan54^\circ\]
= RHS
Hence proved .
APPEARS IN
संबंधित प्रश्न
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:
`cos ((3pi)/ 2 + x ) cos(2pi + x) [cot ((3pi)/2 - x) + cot (2pi + x)]= 1`
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove the following:
`(sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x`
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
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 \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
cos (A − B)
If \[\sin A = \frac{3}{5}, \cos B = - \frac{12}{13}\], where A and B both lie in second quadrant, find the value of 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:
cos (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).
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:
sin 78° cos 18° − cos 78° sin 18°
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:
sin (A + B)
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:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
Prove that:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
If tan A = x tan B, prove that
\[\frac{\sin \left( A - B \right)}{\sin \left( A + B \right)} = \frac{x - 1}{x + 1}\]
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
Prove that:
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
Write the maximum value of 12 sin x − 9 sin2 x.
If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.
If A + B = C, then write the value of tan A tan B tan C.
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
If cot (α + β) = 0, sin (α + 2β) is equal to
If tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
Express the following as the sum or difference of sines and cosines:
2 sin 3x cos x
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 tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
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 tanθ + tan2θ + `sqrt(3)` tanθ tan2θ = `sqrt(3)`, then θ = `("n"pi)/3 + pi/9`
