Advertisements
Advertisements
प्रश्न
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
Advertisements
उत्तर
\[\text{ LHS }\hspace{0.167em} = 2\cot(\alpha - \beta)\]
\[ = \frac{2(1 + \tan\alpha\tan\beta)}{\left[ \tan\alpha - \tan\beta \right]}\]
\[= \frac{2 + 2\left( x + 1 \right)\left( x - 1 \right)}{\left( x + 1 - x + 1 \right)}\]
\[ = \frac{2 + 2 x^2 - 2}{2}\]
\[ = \frac{2 x^2}{2}\]
\[ = x^2 \]
= RHS
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that `2 sin^2 pi/6 + cosec^2 (7pi)/6 cos^2 pi/3 = 3/2`
Find the value of: tan 15°
Prove the following:
`cos ((3pi)/ 2 + x ) cos(2pi + x) [cot ((3pi)/2 - x) + cot (2pi + x)]= 1`
Prove the following:
`(sin x + sin 3x)/(cos x + cos 3x) = tan 2x`
Prove the following:
cos 4x = 1 – 8sin2 x cos2 x
Prove that: `((sin 7x + sin 5x) + (sin 9x + sin 3x))/((cos 7x + cos 5x) + (cos 9x + cos 3x)) = tan 6x`
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 \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
sin (A + B)
Prove that:
Prove that:
If \[\tan A = \frac{5}{6}\text{ and }\tan B = \frac{1}{11}\], prove that \[A + B = \frac{\pi}{4}\].
If \[\tan A = \frac{m}{m - 1}\text{ and }\tan B = \frac{1}{2m - 1}\], then prove that \[A - B = \frac{\pi}{4}\].
Prove that:
cos2 A + cos2 B − 2 cos A cos B cos (A + B) = sin2 (A + B)
If tan A = x tan B, prove that
\[\frac{\sin \left( A - B \right)}{\sin \left( A + B \right)} = \frac{x - 1}{x + 1}\]
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 tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) \[\frac{1}{a} - \frac{1}{b}\].
Prove that:
If angle \[\theta\] is divided into two parts such that the tangents of one part is \[\lambda\] times the tangent of other, and \[\phi\] is their difference, then show that\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
Find the maximum and minimum values of each of the following trigonometrical expression:
\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]
Show that sin 100° − sin 10° is positive.
Write the maximum value of 12 sin x − 9 sin2 x.
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
If \[\frac{\cos \left( x - y \right)}{\cos \left( x + y \right)} = \frac{m}{n}\] then write the value of tan x tan y.
If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
If tan (A − B) = 1 and sec (A + B) = \[\frac{2}{\sqrt{3}}\], the smallest positive value of B is
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
The maximum value of \[\sin^2 \left( \frac{2\pi}{3} + x \right) + \sin^2 \left( \frac{2\pi}{3} - x \right)\] is
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.
Match each item given under column C1 to its correct answer given under column C2.
| C1 | C2 |
| (a) `(1 - cosx)/sinx` | (i) `cot^2 x/2` |
| (b) `(1 + cosx)/(1 - cosx)` | (ii) `cot x/2` |
| (c) `(1 + cosx)/sinx` | (iii) `|cos x + sin x|` |
| (d) `sqrt(1 + sin 2x)` | (iv) `tan x/2` |
If tanθ = `(sinalpha - cosalpha)/(sinalpha + cosalpha)`, then show that sinα + cosα = `sqrt(2)` cosθ.
[Hint: Express tanθ = `tan (alpha - pi/4) theta = alpha - pi/4`]
If sinθ + cosθ = 1, then find the general value of θ.
Find the most general value of θ satisfying the equation tan θ = –1 and cos θ = `1/sqrt(2)`.
If cotθ + tanθ = 2cosecθ, then find the general value of θ.
If f(x) = cos2x + sec2x, then ______.
[Hint: A.M ≥ G.M.]
