Advertisements
Advertisements
Question
Prove that:
Solution
Consider LHS:
\[ \frac{\sin A + \sin 3A + \sin 5A}{\cos A + \cos 3A + \cos 5A}\]
\[ = \frac{\sin A + \sin 5A + \sin 3A}{\cos A + \cos 5A + \cos 3A}\]
\[ = \frac{2\sin \left( \frac{A + 5A}{2} \right) \cos \left( \frac{A - 5A}{2} \right) + \sin 3A}{2\cos \left( \frac{A + 5A}{2} \right) \cos \left( \frac{A - 5A}{2} \right) + \cos 3A}\]
\[ \]
\[ = \frac{2 \sin 3A \cos \left( - 2A \right) + \sin 3A}{2\cos 3A \cos \left( - 2A \right) + \cos 3A}\]
\[ = \frac{2\sin 3A \cos 2A + \sin 3A}{2\cos 3A \cos 2A + \cos 3A}\]
\[ = \frac{\sin 3A \left[ 2\cos 2A + 1 \right]}{\cos 3A \left[ 2\cos 2A + 1 \right]}\]
\[ = \tan 3A\]
= RHS
Hence, RHS = LHS .
APPEARS IN
RELATED QUESTIONS
Prove that:
Show that :
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
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 38° + sin 22° = sin 82°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 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:
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:
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 \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], 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.
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
sin 47° + sin 61° − sin 11° − sin 25° is equal to
If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =
If \[\tan\alpha = \frac{x}{x + 1}\] and
Express the following as the sum or difference of sine or cosine:
cos(60° + A) sin(120° + A)
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
Evaluate:
sin 50° – sin 70° + sin 10°
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)]`
