Advertisements
Advertisements
Question
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Advertisements
Solution
we know that,
cos A - cos B = -2 sin `((A + B)/2) sin((A - B)/2), sin A - sin B = 2 cos ((A + B)/2) sin ((A - B)/2)`
L.H.S. = `(cos9x - cos5x)/(sin17x - sin 3x)`
= `(-2sin ((9x + 5x)/2) sin ((9x - 5x)/2))/(2cos((17x+3x)/2) sin((17x - 3x)/2))`
= `(-2sin7xsin2x)/(2cos10xsin7x)`
= `(-sin2x)/(cos10x)` = R.H.S.
APPEARS IN
RELATED QUESTIONS
Prove the following:
sin 2x + 2sin 4x + sin 6x = 4cos2 x sin 4x
Prove the following:
`(sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x`
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove the following:
cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
cos (A − B)
If \[\sin A = \frac{3}{5}, \cos B = - \frac{12}{13}\], where A and B both lie in second quadrant, find the value of sin (A + B).
Evaluate the following:
cos 80° cos 20° + sin 80° sin 20°
If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
tan (A + B)
Prove that
\[\frac{\tan A + \tan B}{\tan A - \tan B} = \frac{\sin \left( A + B \right)}{\sin \left( A - B \right)}\]
Prove that:
If \[\tan A = \frac{5}{6}\text{ and }\tan B = \frac{1}{11}\], prove that \[A + B = \frac{\pi}{4}\].
Prove that:
Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
Prove that sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
If x lies in the first quadrant and \[\cos x = \frac{8}{17}\], then prove that:
If tan x + \[\tan \left( x + \frac{\pi}{3} \right) + \tan \left( x + \frac{2\pi}{3} \right) = 3\], then prove that \[\frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x} = 1\].
If α, β are two different values of x lying between 0 and 2π, which satisfy the equation 6 cos x + 8 sin x = 9, find the value of sin (α + β).
Find the maximum and minimum values of each of the following trigonometrical expression:
12 cos x + 5 sin x + 4
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
Reduce each of the following expressions to the sine and cosine of a single expression:
cos x − sin x
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
Show that sin 100° − sin 10° is positive.
If α + β − γ = π and sin2 α +sin2 β − sin2 γ = λ sin α sin β cos γ, then write the value of λ.
If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
If A + B + C = π, then \[\frac{\tan A + \tan B + \tan C}{\tan A \tan B \tan C}\] is equal to
Find the most general value of θ satisfying the equation tan θ = –1 and cos θ = `1/sqrt(2)`.
If cos(θ + Φ) = m cos(θ – Φ), then prove that 1 tan θ = `(1 - m)/(1 + m) cot phi`
[Hint: Express `(cos(theta + Φ))/(cos(theta - Φ)) = m/1` and apply Componendo and Dividendo]
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
The value of tan3A - tan2A - tanA is equal to ______.
The value of `cot(pi/4 + theta)cot(pi/4 - theta)` is ______.
If sinθ + cosθ = 1, then the value of sin2θ is equal to ______.
If sinx + cosx = a, then sin6x + cos6x = ______.
If sinx + cosx = a, then |sinx – cosx| = ______.
State whether the statement is True or False? Also give justification.
If tanA = `(1 - cos B)/sinB`, then tan2A = tanB
