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
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
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:
cos 12x - cos 4x
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
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:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
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 \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
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.
The value of sin 50° − sin 70° + sin 10° is equal to
sin 47° + sin 61° − sin 11° − sin 25° is equal to
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:
tan 20° tan 40° tan 80° = `sqrt3`.
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 sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.
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:
Find the value of tan22°30′. `["Hint:" "Let" θ = 45°, "use" tan theta/2 = (sin theta/2)/(cos theta/2) = (2sin theta/2 cos theta/2)/(2cos^2 theta/2) = sintheta/(1 + costheta)]`
