Advertisements
Advertisements
प्रश्न
Prove that:
cos 20° + cos 100° + cos 140° = 0
Advertisements
उत्तर
Consider LHS:
\[\cos 20^\circ + \cos 100^\circ + \cos 140^\circ\]
\[ = 2\cos \left( \frac{20^\circ + 100^\circ}{2} \right) \cos \left( \frac{20^\circ - 100^\circ}{2} \right) + \cos 140^\circ \left\{ \because \cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\cos 60^\circ \cos \left( - 40^\circ \right) + \cos 140^\circ\]
\[ = 2 \times \frac{1}{2}\cos 40^\circ + \cos 140^\circ\]
\[ = \cos 40^\circ+ \cos 140^\circ\]
\[ = 2\cos \left( \frac{40^\circ + 140^\circ}{2} \right) \cos \left( \frac{40^\circ - 140^\circ}{2} \right)\]
\[ = 2 \cos 90^\circ \cos 50^\circ\]
\[ = 0\]
Hence, LHS = RHS.
APPEARS IN
संबंधित प्रश्न
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 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 5x − sin x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
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 \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
cos 40° + cos 80° + cos 160° + cos 240° =
sin 163° cos 347° + sin 73° sin 167° =
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
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 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 6θ – sin 2θ
Prove that:
cos 20° cos 40° cos 80° = `1/8`
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
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.
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:
