Advertisements
Advertisements
प्रश्न
Prove that:
cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
Advertisements
उत्तर
Consider LHS:
\[ \cos A + \cos 3A + \cos 5A + \cos 7A\]
\[ = 2\cos \left( \frac{A + 3A}{2} \right) \cos \left( \frac{A - 3A}{2} \right) + 2\cos \left( \frac{5A + 7A}{2} \right) \cos \left( \frac{5A - 7A}{2} \right) \left\{ \because \cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\cos 2A \cos\left( - A \right) + 2\cos 6A \cos\left( - A \right)\]
\[= 2\cos 2A \cos A + 2\cos 6A \cos A\]
\[ = 2\cos A(\cos 2A + \cos 6A)\]
\[ = 2\cos A \times 2\cos \left( \frac{2A + 6A}{2} \right) \cos \left( \frac{2A - 6A}{2} \right)\]
\[ = 4\cos A \cos 4A \cos\left( - 2A \right)\]
\[ = 4\cos A \cos 2A \cos 4A\]
= RHS
Hence, LHS = RHS.
APPEARS IN
संबंधित प्रश्न
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
Express each of the following as the product of sines and cosines:
cos 12x - cos 4x
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
cos 55° + cos 65° + cos 175° = 0
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:
Prove that:
Prove that:
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
sin 163° cos 347° + sin 73° sin 167° =
If sin 2 θ + sin 2 ϕ = \[\frac{1}{2}\] and cos 2 θ + cos 2 ϕ = \[\frac{3}{2}\], then cos2 (θ − ϕ) =
The value of sin 50° − sin 70° + sin 10° is equal to
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
Express the following as the sum or difference of sine or cosine:
cos(60° + A) sin(120° + A)
Express the following as the sum or difference of sine or cosine:
`cos (7"A")/3 sin (5"A")/3`
Express the following as the sum or difference of sine or cosine:
cos 7θ sin 3θ
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Evaluate-
cos 20° + cos 100° + cos 140°
