Advertisements
Advertisements
Question
If tanθ = `a/b`, then bcos2θ + asin2θ is equal to ______.
Options
a
b
`a/b`
None
Advertisements
Solution
If tanθ = `a/b`, then bcos2θ + asin2θ is equal to b.
Explanation:
Given that: tanθ = `a/b`
bcos2θ + asin2θ = `b[(1 - tan^2 theta)/(1 + tan^2 theta)] + a[(2 tan theta)/(1 + tan^2 theta)]`
= `b[(1 - a^2/b^2)/(1 + a^2/b^2)] + a[((2a)/b)/(1 + a^2/b^2)]`
= `b[(b^2 - a^2)/(b^2 + a^2)] + [((2a^2)/b)/((b^2 + a^2)/b^2)]`
= `(b^3 - a^2b)/(b^2 + a^2) + (2a^2b)/(b^2 + a^2)`
= `(b^3 - a^2b + 2a^2b)/(b^2 + a^2)`
= `(b^3 + a^2b)/(b^2 + a^2)`
= `(b(b^2 + a^2))/(b^2 + a^2)`
= b
APPEARS IN
RELATED QUESTIONS
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:
sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
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:
cos (A − B)
If \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
cos (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:
sin (A + B)
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:
\[\frac{\sin \left( A - B \right)}{\cos A \cos B} + \frac{\sin \left( B - C \right)}{\cos B \cos C} + \frac{\sin \left( C - A \right)}{\cos C \cos A} = 0\]
Prove that:
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
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 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).
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.
If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).
Find the maximum and minimum values of each of the following trigonometrical expression:
12 sin x − 5 cos x
Reduce each of the following expressions to the sine and cosine of a single expression:
\[\sqrt{3} \sin x - \cos x\]
Show that sin 100° − sin 10° is positive.
Write the maximum value of 12 sin x − 9 sin2 x.
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =
tan 3A − tan 2A − tan A =
If cot (α + β) = 0, sin (α + 2β) is equal to
If tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
Express the following as the sum or difference of sines and cosines:
2 sin 3x cos x
Express the following as the sum or difference of sines and cosines:
2 cos 7x cos 3x
Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α
If sinθ + cosθ = 1, then find the general value of θ.
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`]
The value of sin(45° + θ) - cos(45° - θ) is ______.
The value of `cot(pi/4 + theta)cot(pi/4 - theta)` is ______.
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.
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`
