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 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Express each of the following as the product of sines and cosines:
sin 12x + sin 4x
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 8x
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:
cos 100° + cos 20° = cos 40°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
Prove that:
cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
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 \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], 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 α + cos β)2 + (sin α + sin β)2 = \[\lambda \cos^2 \left( \frac{\alpha - \beta}{2} \right)\], write the value of λ.
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
If \[\tan\alpha = \frac{x}{x + 1}\] and
Express the following as the sum or difference of sine or cosine:
`sin "A"/8 sin (3"A")/8`
Express the following as the sum or difference of sine or cosine:
cos 7θ sin 3θ
Prove that:
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0
Evaluate:
sin 50° – sin 70° + sin 10°
