Advertisements
Advertisements
प्रश्न
Prove that:
Advertisements
उत्तर
Consider LHS:
\[ \frac{\sin 11A \sin A + \sin 7A \sin 3A}{\cos 2A \sin A + \cos 6A \sin 3A}\]
Multiplying numerator and denominator by 2, we get
\[ = \frac{2\sin 11A \sin A + 2\sin 7A \sin 3A}{2\cos 11A sin A + 2\cos 7A \sin 3A}\]
\[ = \frac{\cos \left( 11A - A \right) - \cos \left( 11A + A \right) + \cos \left( 7A - 3A \right) - \cos \left( 7A + 3A \right)}{\sin \left( 11A + A \right) - \sin \left( 11A - A \right) + \sin \left( 7A + 3A \right) - \sin \left( 7A - 3A \right)}\]
\[ = \frac{\cos 10A - \cos 12A + \cos 4A - \cos 10A}{\sin 12A - \sin 10A + \sin 10A - \sin 4A}\]
\[ = \frac{\cos 4A - \cos 12A}{\sin 12A - \sin 4A}\]
\[ = \frac{- 2\sin \left( \frac{4A + 12A}{2} \right) \sin \left( \frac{4A - 12A}{2} \right)}{2\sin \left( \frac{12A - 4A}{2} \right) \cos \left( \frac{12A + 4A}{2} \right)}\]
\[ = \frac{- \sin 8A \sin \left( - 4A \right)}{\sin 4A \cos 8A}\]
\[ = \frac{\sin 8A \sin 4A}{\sin 4A \cos 8A}\]
\[ = \tan8A\]
= RHS
Hence, LHS = RHS.
APPEARS IN
संबंधित प्रश्न
Prove that:
Prove that:
Prove that:
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
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:
sin 12x + sin 4x
Express each of the following as the product of sines and cosines:
cos 12x + cos 8x
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
Prove that:
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
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:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
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 sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
cos 35° + cos 85° + cos 155° =
If \[\tan\alpha = \frac{x}{x + 1}\] and
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 product of sine and cosine.
sin A + sin 2A
Express the following as the product of sine and cosine.
cos 2A + cos 4A
Prove that:
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.
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)]`
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
