Advertisements
Advertisements
प्रश्न
If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
विकल्प
(a) 0
(b) −1
(c) 1
(d) None of these
उत्तर
(b) −1
π = 180°
Now, using the identities
\[\cos\left( A + B \right) = \cos A\cos B - \sin A\sin B\] and \[\sin\left( A + B \right) = \sin A\cos B + \cos A\sin B\]
\[\sec A\left( \cos B\cos C - \sin B\sin C \right) = \frac{- \cos A\cos B^2 + \cos B\sin A\sin B - \sin B\sin A\cos B - \sin^2 B\cos A}{\cos A}\]
\[\Rightarrow \sec A\left( \cos B\cos C - \sin B\sin C \right) = \frac{- \cos A\left( \cos^2 B + \sin^2 B \right)}{\cos A}\]
\[ \Rightarrow \sec A\left( \cos B\cos C - \sin B\sin C \right) = \frac{- \cos A}{\cos A} = - 1\]
APPEARS IN
संबंधित प्रश्न
Prove that: `2 sin^2 (3pi)/4 + 2 cos^2 pi/4 + 2 sec^2 pi/3 = 10`
Find the value of: tan 15°
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 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x`
Prove the following:
cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
Prove the following:
cos 6x = 32 cos6 x – 48 cos4 x + 18 cos2 x – 1
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 \[\sin A = \frac{3}{5}, \cos B = - \frac{12}{13}\], 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 \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], where π < A < \[\frac{3\pi}{2}\] and 0 < B <\[\frac{\pi}{2}\], find tan (A + B).
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
Prove that:
\[\tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} = 1\]
Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
Prove that sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
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 sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
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.
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 \[\cos P = \frac{1}{7}\text{ and }\cos Q = \frac{13}{14}\], where P and Q both are acute angles. Then, the value of P − Q is
If \[\tan\theta = \frac{1}{2}\] and \[\tan\phi = \frac{1}{3}\], then the value of \[\tan\phi = \frac{1}{3}\] is
If tan (π/4 + x) + tan (π/4 − x) = a, then tan2 (π/4 + x) + tan2 (π/4 − x) =
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
If cotθ + tanθ = 2cosecθ, then find the general value of θ.
If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a2 - 2b2
[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]
If f(x) = cos2x + sec2x, then ______.
[Hint: A.M ≥ G.M.]
The value of tan 75° - cot 75° is equal to ______.
The value of tan3A - tan2A - tanA is equal to ______.
The value of `cot(pi/4 + theta)cot(pi/4 - theta)` is ______.
If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.
If tanθ = `a/b`, then bcos2θ + asin2θ is equal to ______.
If sinx + cosx = a, then |sinx – cosx| = ______.
