Advertisements
Advertisements
Question
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Advertisements
Solution
\[LHS = \sin\left( \frac{5\pi}{18} \right) - \cos\frac{4\pi}{9}\]
\[ = \sin\left( \frac{5\pi}{18} \right) - \cos\left( \frac{\pi}{2} - \frac{\pi}{18} \right)\]
\[ = \sin\left( \frac{5\pi}{18} \right) - \sin\left( \frac{\pi}{18} \right)\]
\[ = 2\sin\left( \frac{\frac{5\pi}{18} - \frac{\pi}{18}}{2} \right)\cos\left( \frac{\frac{5\pi}{18} + \frac{\pi}{18}}{2} \right) \left[ \because \sin A - \sin B = 2\sin\left( \frac{A - B}{2} \right)\cos\left( \frac{A + B}{2} \right) \right]\]
\[ = 2\sin\left( \frac{\pi}{9} \right)\cos\frac{\pi}{6}\]
\[ = 2\sin\left( \frac{\pi}{9} \right)\cos\frac{\pi}{6}\]
\[ = 2 \times \frac{\sqrt{3}}{2}\sin\left( \frac{\pi}{9} \right)\]
\[ = \sqrt{3}\sin\left( \frac{\pi}{9} \right) = RHS\]
Hence, LHS = RHS.
APPEARS IN
RELATED QUESTIONS
Prove that:
Prove that:
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 (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Express each of the following as the product of sines and cosines:
cos 12x - cos 4x
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
Prove that:
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 cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
Prove that:
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
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 cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
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° =
cos 35° + cos 85° + cos 155° =
Express the following as the sum or difference of sine or cosine:
`sin "A"/8 sin (3"A")/8`
Express the following as the product of sine and cosine.
sin A + sin 2A
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
Evaluate:
sin 50° – sin 70° + sin 10°
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.
