Advertisements
Advertisements
प्रश्न
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
Advertisements
उत्तर
Consider LHS:
\[ \cos 3A + cos 5A + \cos 7A + \cos 15A\]
\[ = 2\cos \left( \frac{3A + 5A}{2} \right) \cos \left( \frac{3A - 5A}{2} \right) + 2\cos \left( \frac{7A + 15A}{2} \right) \cos \left( \frac{7A - 15A}{2} \right) \left\{ \because \cos A + \cos B = 2\cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\cos 4A \cos\left( - A \right) + 2\cos 11A \cos\left( - 4A \right)\]
\[= 2\cos 4A \cos A + 2\cos 11A \cos 4A\]
\[ = 2\cos 4A \left\{ \cos A + \cos 11A \right\}\]
\[ = 2\cos 4A \times \left\{ 2\cos \left( \frac{A + 11A}{2} \right) \cos \left( \frac{A - 11A}{2} \right) \right\}\]
\[ = 4\cos 4A \cos 6A \cos\left( - 5A \right)\]
\[ = 4\cos 4A \cos 5A \cos 6A\]
= RHS
Hence, LHS = RHS
APPEARS IN
संबंधित प्रश्न
Show that :
Express each of the following as the product of sines and cosines:
cos 12x + cos 8x
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
Prove that:
Prove that:
cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
Prove that:
Prove that:
Prove that:
Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
Prove that:
Prove that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 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 A + B = \[\frac{\pi}{3}\] and cos A + cos B = 1, then find the value of cos \[\frac{A - B}{2}\].
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
sin 163° cos 347° + sin 73° sin 167° =
The value of cos 52° + cos 68° + cos 172° is
sin 47° + sin 61° − sin 11° − sin 25° is equal to
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
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 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 7θ sin 3θ
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
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.
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
