Advertisements
Advertisements
प्रश्न
Express the following as the sum or difference of sines and cosines:
2 cos 7x cos 3x
उत्तर
\[2\left( \cos 7x \right) \left( \cos 3x \right)\]
\[ = \cos \left( 7x + 3x \right) + \cos \left( 7x - 3x \right) \left[ \because 2 \cos A \cos B = \cos(A + B) + \cos(A - B) \right]\]
\[ = \cos 10x + \cos 4x\]
APPEARS IN
संबंधित प्रश्न
Prove the following: `(tan(pi/4 + x))/(tan(pi/4 - x)) = ((1+ tan x)/(1- tan x))^2`
Prove the following:
`(cos (pi + x) cos (-x))/(sin(pi - x) cos (pi/2 + x)) = cot^2 x`
Prove the following:
`tan 4x = (4tan x(1 - tan^2 x))/(1 - 6tan^2 x + tan^4 x)`
Prove the following:
cos 4x = 1 – 8sin2 x cos2 x
Prove that: `((sin 7x + sin 5x) + (sin 9x + sin 3x))/((cos 7x + cos 5x) + (cos 9x + cos 3x)) = tan 6x`
Prove that: sin 3x + sin 2x – sin x = 4sin x `cos x/2 cos (3x)/2`
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 \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], where π < A < \[\frac{3\pi}{2}\] and 0 < B <\[\frac{\pi}{2}\], find tan (A + B).
If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
sin (A + B)
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)
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:
tan (A + B)
Prove that:
Prove that:
sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
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:
\[\frac{\tan \left( A + B \right)}{\cot \left( A - B \right)} = \frac{\tan^2 A - \tan^2 B}{1 - \tan^2 A \tan^2 B}\]
Prove that:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
Find the maximum and minimum values of each of the following trigonometrical expression:
12 sin x − 5 cos x
Find the maximum and minimum values of each of the following trigonometrical expression:
12 cos x + 5 sin x + 4
Find the maximum and minimum values of each of the following trigonometrical expression:
\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
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).
tan 20° + tan 40° + \[\sqrt{3}\] tan 20° tan 40° is equal to
If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
If tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
If sin (π cos x) = cos (π sin x), then sin 2x = ______.
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.
Match each item given under column C1 to its correct answer given under column C2.
C1 | C2 |
(a) `(1 - cosx)/sinx` | (i) `cot^2 x/2` |
(b) `(1 + cosx)/(1 - cosx)` | (ii) `cot x/2` |
(c) `(1 + cosx)/sinx` | (iii) `|cos x + sin x|` |
(d) `sqrt(1 + sin 2x)` | (iv) `tan x/2` |
If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].
If f(x) = cos2x + sec2x, then ______.
[Hint: A.M ≥ G.M.]
The value of tan 75° - cot 75° is equal to ______.
If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.
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`