Advertisements
Advertisements
प्रश्न
Reduce each of the following expressions to the sine and cosine of a single expression:
cos x − sin x
उत्तर
\[\text{ Let } f\left( x \right) = \cos x - \sin x\]
\[\text{ Dividing and multiplying by } \sqrt{1^2 + 1^2}, i . e . \text{ by }\sqrt{2,} \text{ we get } : \]
\[ f\left( x \right) = \sqrt{2}\left( \frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x \right)\]
\[ \Rightarrow f\left( x \right) = \sqrt{2}(\cos45°\cos x - \sin45°\sin x) \]
\[ \Rightarrow f\left( x \right) = \sqrt{2}\cos\left( \frac{\pi}{4} + x \right)\]
\[\text{ Again }, \]
\[ f\left( x \right) = \sqrt{2}\left( \frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x \right)\]
\[ \Rightarrow f\left( x \right) = \sqrt{2}(\sin45°\cos x - \cos45∏\sin x)\]
\[ \Rightarrow f(x) = \sqrt{2} \sin\left( \frac{\pi}{4} - x \right)\]
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:
sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x
Prove the following:
`(sin x - sin 3x)/(sin^2 x - cos^2 x) = 2sin x`
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:
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 \[\sin A = \frac{12}{13}\text{ and } \sin B = \frac{4}{5}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
sin (A + B)
If \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
cos (A + B)
If \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], 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).
Evaluate the following:
sin 78° cos 18° − cos 78° sin 18°
Prove that:
\[\frac{7\pi}{12} + \cos\frac{\pi}{12} = \sin\frac{5\pi}{12} - \sin\frac{\pi}{12}\]
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:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
Prove that:
cos2 A + cos2 B − 2 cos A cos B cos (A + B) = sin2 (A + B)
Prove that sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
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 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 α, β are two different values of x lying between 0 and 2π, which satisfy the equation 6 cos x + 8 sin x = 9, find the value of sin (α + β).
If angle \[\theta\] is divided into two parts such that the tangents of one part is \[\lambda\] times the tangent of other, and \[\phi\] is their difference, then show that\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).
Prove that \[\left( 2\sqrt{3} + 3 \right) \sin x + 2\sqrt{3} \cos x\] lies between \[- \left( 2\sqrt{3} + \sqrt{15} \right) \text{ and } \left( 2\sqrt{3} + \sqrt{15} \right)\]
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
If \[\frac{\cos \left( x - y \right)}{\cos \left( x + y \right)} = \frac{m}{n}\] then write the value of tan x tan y.
If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.
The value of \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is
If tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
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
Express the following as the sum or difference of sines and cosines:
2 sin 4x sin 3x
If cotθ + tanθ = 2cosecθ, then find the general value of θ.
Find the general solution of the equation `(sqrt(3) - 1) costheta + (sqrt(3) + 1) sin theta` = 2
[Hint: Put `sqrt(3) - 1` = r sinα, `sqrt(3) + 1` = r cosα which gives tanα = `tan(pi/4 - pi/6)` α = `pi/12`]
If sinθ + cosθ = 1, then the value of sin2θ is equal to ______.
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.
