Advertisements
Advertisements
Question
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
Advertisements
Solution
\[\text{ RHS }= \sin^2 A + \sin^2 \left( A - B \right) - 2\sin A \cos B \sin\left( A - B \right)\]
\[ = \sin^2 A + \sin\left( A - B \right) \left\{ \sin\left( A - B \right) - 2\sin A \cos B \right\}\]
\[ = \sin^2 A + \sin\left( A - B \right) \left( \sin A \cos B - \cos A \sin B - 2\sin A \cos B \right)\]
\[ = \sin^2 A + \sin\left( A - B \right) \left( - \sin A \cos B - \cos A \sin B \right)\]
\[ = \sin^2 A - \sin\left( A - B \right) \left( \sin A \cos B + \cos A \sin B \right)\]
\[ = \sin^2 A - \sin\left( A - B \right) \sin\left( A + B \right)\]
\[ = \sin^2 A - \left( \sin^2 A - \sin^2 B \right)\]
\[ = \sin^2 A - \sin^2 A + \sin^2 B\]
\[ = \sin^2 B\]
= LHS
Hence proved.
APPEARS IN
RELATED QUESTIONS
Prove that `2 sin^2 pi/6 + cosec^2 (7pi)/6 cos^2 pi/3 = 3/2`
Prove that: `2 sin^2 (3pi)/4 + 2 cos^2 pi/4 + 2 sec^2 pi/3 = 10`
Find the value of: sin 75°
Find the value of: tan 15°
Prove the following: `cos (pi/4 xx x) cos (pi/4 - y) - sin (pi/4 - x)sin (pi/4 - y) = sin (x + y)`
Prove the following:
`cos ((3pi)/4 + x) - cos((3pi)/4 - x) = -sqrt2 sin x`
Prove the following:
`(sin x - sin 3x)/(sin^2 x - cos^2 x) = 2sin x`
Prove that: `((sin 7x + sin 5x) + (sin 9x + sin 3x))/((cos 7x + cos 5x) + (cos 9x + cos 3x)) = tan 6x`
Prove that: sin 3x + sin 2x – sin x = 4sin x `cos x/2 cos (3x)/2`
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).
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A - B)
Evaluate the following:
cos 47° cos 13° − sin 47° sin 13°
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:
\[\frac{7\pi}{12} + \cos\frac{\pi}{12} = \sin\frac{5\pi}{12} - \sin\frac{\pi}{12}\]
Prove that:
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
If sin (α + β) = 1 and sin (α − β) \[= \frac{1}{2}\], where 0 ≤ α, \[\beta \leq \frac{\pi}{2}\], then find the values of tan (α + 2β) and tan (2α + β).
If sin α + sin β = a and cos α + cos β = b, show that
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
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:
\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]
Reduce each of the following expressions to the sine and cosine of a single expression:
\[\sqrt{3} \sin x - \cos x\]
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
Write the interval in which the value of 5 cos x + 3 cos \[\left( x + \frac{\pi}{3} \right) + 3\] lies.
If tan (A + B) = p and tan (A − B) = q, then write the value of tan 2B.
If tan \[\alpha = \frac{1}{1 + 2^{- x}}\] and \[\tan \beta = \frac{1}{1 + 2^{x + 1}}\] then write the value of α + β lying in the interval \[\left( 0, \frac{\pi}{2} \right)\]
tan 20° + tan 40° + \[\sqrt{3}\] tan 20° tan 40° is equal to
If \[\tan\alpha = \frac{x}{x + 1}\] and \[\tan\alpha = \frac{x}{x + 1}\], then \[\alpha + \beta\] is equal to
Express the following as the sum or difference of sines and cosines:
2 sin 4x sin 3x
If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].
If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a2 - 2b2
[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]
Find the general solution of the equation `(sqrt(3) - 1) costheta + (sqrt(3) + 1) sin theta` = 2
[Hint: Put `sqrt(3) - 1` = r sinα, `sqrt(3) + 1` = r cosα which gives tanα = `tan(pi/4 - pi/6)` α = `pi/12`]
If f(x) = cos2x + sec2x, then ______.
[Hint: A.M ≥ G.M.]
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
If sinx + cosx = a, then |sinx – cosx| = ______.
State whether the statement is True or False? Also give justification.
If tanθ + tan2θ + `sqrt(3)` tanθ tan2θ = `sqrt(3)`, then θ = `("n"pi)/3 + pi/9`
