Advertisements
Advertisements
प्रश्न
If \[\tan\alpha = \frac{x}{x + 1}\] and \[\tan\alpha = \frac{x}{x + 1}\], then \[\alpha + \beta\] is equal to
विकल्प
- \[\frac{\pi}{2}\]
- \[\frac{\pi}{3}\]
- \[\frac{\pi}{6}\]
- \[\frac{\pi}{4}\]
उत्तर
It is given that \[\tan\alpha = \frac{x}{x + 1}\] and
\[\tan\left( \alpha + \beta \right) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}\]
\[ = \frac{\frac{x}{x + 1} + \frac{1}{2x + 1}}{1 - \frac{x}{x + 1} \times \frac{1}{2x + 1}}\]
\[ = \frac{\frac{x\left( 2x + 1 \right) + \left( x + 1 \right)}{\left( x + 1 \right)\left( 2x + 1 \right)}}{\frac{\left( x + 1 \right)\left( 2x + 1 \right) - x}{\left( x + 1 \right)\left( 2x + 1 \right)}}\]
\[ = \frac{2 x^2 + x + x + 1}{2 x^2 + 3x + 1 - x}\]
\[= \frac{2 x^2 + 2x + 1}{2 x^2 + 2x + 1}\]
\[ = 1\]
\[\therefore \alpha + \beta = \frac{\pi}{4} \left( \tan\frac{\pi}{4} = 1 \right)\]
Hence, the correct answer is option D.
APPEARS IN
संबंधित प्रश्न
Prove the following:
cos2 2x – cos2 6x = sin 4x sin 8x
Prove the following:
`(sin x - sin 3x)/(sin^2 x - cos^2 x) = 2sin x`
Prove the following:
cos 4x = 1 – 8sin2 x cos2 x
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:
sin (A − B)
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 \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], 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°
If \[\tan A = \frac{5}{6}\text{ and }\tan B = \frac{1}{11}\], prove that \[A + B = \frac{\pi}{4}\].
Prove that: \[\frac{\sin \left( A + B \right) + \sin \left( A - B \right)}{\cos \left( A + B \right) + \cos \left( A - B \right)} = \tan A\]
Prove that:
cos2 A + cos2 B − 2 cos A cos B cos (A + B) = sin2 (A + B)
Prove that:
\[\frac{\tan \left( A + B \right)}{\cot \left( A - B \right)} = \frac{\tan^2 A - \tan^2 B}{1 - \tan^2 A \tan^2 B}\]
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
Prove that sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
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 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 sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
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:
\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]
Reduce each of the following expressions to the sine and cosine of a single expression:
\[\sqrt{3} \sin x - \cos x\]
Reduce each of the following expressions to the sine and cosine of a single expression:
cos x − sin x
Write the maximum value of 12 sin x − 9 sin2 x.
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 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 =
If A + B + C = π, then \[\frac{\tan A + \tan B + \tan C}{\tan A \tan B \tan C}\] is equal to
If cot (α + β) = 0, sin (α + 2β) is equal to
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
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
If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].
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 tanα = `m/(m + 1)`, tanβ = `1/(2m + 1)`, then α + β 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 cosecx = 1 + cotx then x = 2nπ, 2nπ + `pi/2`
