Advertisements
Advertisements
प्रश्न
Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
Advertisements
उत्तर
\[\cos^2 45^\circ - \sin^2 15^\circ\]
\[ = \cos\left( 45^\circ + 15^\circ \right)\cos\left( 45^\circ - 15^\circ \right) \left[ \cos^2 X - \sin^2 Y = \cos\left( X + Y \right)\cos\left( X - Y \right) \right]\]
\[ = \cos60^\circ\cos30^\circ\]
\[ = \frac{1}{2} \times \frac{\sqrt{3}}{2}\]
\[ = \frac{\sqrt{3}}{4}\]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that: `2 sin^2 (3pi)/4 + 2 cos^2 pi/4 + 2 sec^2 pi/3 = 10`
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 (pi + x) cos (-x))/(sin(pi - x) cos (pi/2 + x)) = cot^2 x`
Prove the following:
`cos ((3pi)/ 2 + x ) cos(2pi + x) [cot ((3pi)/2 - x) + cot (2pi + x)]= 1`
Prove the following:
sin 2x + 2sin 4x + sin 6x = 4cos2 x sin 4x
Prove the following:
`(sin x - sin 3x)/(sin^2 x - cos^2 x) = 2sin x`
Prove the following:
cos 6x = 32 cos6 x – 48 cos4 x + 18 cos2 x – 1
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 \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
sin (A − B)
If \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
cos (A + B)
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A + B)
Prove that:
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
Prove that:
\[\tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} = 1\]
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
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).
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:
\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
If A + B + C = π, then \[\frac{\tan A + \tan B + \tan C}{\tan A \tan B \tan C}\] is equal to
If \[\cos P = \frac{1}{7}\text{ and }\cos Q = \frac{13}{14}\], where P and Q both are acute angles. Then, the value of P − Q is
The value of \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is
If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
If \[\tan\alpha = \frac{x}{x + 1}\] and \[\tan\alpha = \frac{x}{x + 1}\], then \[\alpha + \beta\] is equal to
Express the following as the sum or difference of sines and cosines:
2 sin 4x sin 3x
Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α
If cotθ + tanθ = 2cosecθ, then find the general value of θ.
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
The value of tan3A - tan2A - tanA is equal to ______.
The value of sin(45° + θ) - cos(45° - θ) is ______.
