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:
Show that :
Prove that:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Express each of the following as the product of sines and cosines:
sin 12x + sin 4x
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
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:
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
Prove that:
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
If \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
If \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
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 \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
cos 35° + cos 85° + cos 155° =
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P., then cot A, cot B and cot Care in
Express the following as the product of sine and cosine.
cos 2A + cos 4A
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0
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 20° cos 40° cos 60° cos 80° = `3/16`.
If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
