Advertisements
Advertisements
प्रश्न
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
Advertisements
उत्तर
\[\text{ Let }x = \cos \alpha \cos \beta\]
\[ \Rightarrow x = \frac{1}{2}\left[ 2\cos \alpha \cos \beta \right]\]
\[ \Rightarrow x = \frac{1}{2}\left[ \cos \left( \alpha + \beta \right) + \cos \left( \alpha - \beta \right) \right]\]
\[ \Rightarrow x = \frac{1}{2}\left[ \cos \left( \alpha - \beta \right) + \cos 90^\circ \right]\]
\[ \Rightarrow x = \frac{1}{2}\cos \left( \alpha - \beta \right)\]
Now,
\[ - 1 \leq \cos \left( \alpha - \beta \right) \leq 1\]
\[ \Rightarrow - \frac{1}{2} \leq \frac{1}{2}\cos\left( \alpha - \beta \right) \leq \frac{1}{2}\]
\[ \Rightarrow - \frac{1}{2} \leq x \leq \frac{1}{2}\]
\[ \Rightarrow - \frac{1}{2} \leq \cos \alpha \cos \beta \leq \frac{1}{2}\]
\[\text{Hence}, \frac{1}{2}\text{ is the maximum value of }\cos \alpha \cos \beta .\]
APPEARS IN
संबंधित प्रश्न
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
Express each of the following as the product of sines and cosines:
sin 5x − sin x
Express each of the following as the product of sines and cosines:
cos 12x - cos 4x
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.
If (cos α + cos β)2 + (sin α + sin β)2 = \[\lambda \cos^2 \left( \frac{\alpha - \beta}{2} \right)\], write the value of λ.
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
cos 40° + cos 80° + cos 160° + cos 240° =
cos 35° + cos 85° + cos 155° =
If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =
Express the following as the product of sine and cosine.
sin A + sin 2A
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
2 cos `pi/13` cos \[\frac{9\pi}{13} + \text{cos} \frac{3\pi}{13} + \text{cos} \frac{5\pi}{13}\] = 0
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`
If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.
