Advertisements
Advertisements
प्रश्न
If tan (π/4 + x) + tan (π/4 − x) = a, then tan2 (π/4 + x) + tan2 (π/4 − x) =
विकल्प
a2 + 1
a2 + 2
a2 − 2
None of these
Advertisements
उत्तर
Given:
\[\tan\left( \frac{\pi}{4} + x \right) + \tan\left( \frac{\pi}{4} - x \right) = a\]
\[ \Rightarrow \left[ \tan\left( \frac{\pi}{4} + x \right) + \tan\left( \frac{\pi}{4} - x \right) \right]^2 = a^2 \]
\[ \Rightarrow \tan^2 \left( \frac{\pi}{4} + x \right) + \tan^2 \left( \frac{\pi}{4} - x \right) + 2 \tan\left( \frac{\pi}{4} - x \right) \tan\left( \frac{\pi}{4} + x \right) = a^2 \]
\[ \Rightarrow \tan^2 \left( \frac{\pi}{4} + x \right) + \tan^2 \left( \frac{\pi}{4} - x \right) = a^2 - 2 \tan\left( \frac{\pi}{4} - x \right) \tan\left( \frac{\pi}{4} + x \right)\]
\[ \Rightarrow \tan^2 \left( \frac{\pi}{4} + x \right) + \tan^2 \left( \frac{\pi}{4} - x \right) = a^2 - 2\left[ \frac{\tan45^\circ - \tan x}{1 + \tan45^\circ \tan x} \times \frac{\tan45^\circ + \tan x}{1 - \tan45^\circ \tan x} \right] \]
\[ \Rightarrow \tan^2 \left( \frac{\pi}{4} + x \right) + \tan^2 \left( \frac{\pi}{4} - x \right) = a^2 - 2\left[ \frac{1^\circ - \tan x}{1 + \tan x} \times \frac{1 + \tan x}{1 - \tan x} \right]\]
\[ \Rightarrow \tan^2 \left( \frac{\pi}{4} + x \right) + \tan^2 \left( \frac{\pi}{4} - x \right) = a^2 - 2\left( \frac{1 - \tan^2 x}{1 - \tan^2 x} \right)\]
\[ \Rightarrow \tan^2 \left( \frac{\pi}{4} + x \right) + \tan^2 \left( \frac{\pi}{4} - x \right) = a^2 - 2\]
APPEARS IN
संबंधित प्रश्न
Prove the following:
`(cos (pi + x) cos (-x))/(sin(pi - x) cos (pi/2 + x)) = cot^2 x`
Prove the following:
`cos ((3pi)/ 2 + x ) cos(2pi + x) [cot ((3pi)/2 - x) + cot (2pi + x)]= 1`
Prove the following:
cos2 2x – cos2 6x = sin 4x sin 8x
If \[\sin A = \frac{12}{13}\text{ and } \sin B = \frac{4}{5}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
sin (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).
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)
If \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], where π < A < \[\frac{3\pi}{2}\] and 0 < B <\[\frac{\pi}{2}\], find tan (A + B).
Evaluate the following:
sin 78° cos 18° − cos 78° sin 18°
Evaluate the following:
sin 36° cos 9° + cos 36° sin 9°
Prove that
Prove that:
Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
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:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
Prove that:
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
If \[\tan\theta = \frac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}\] , then show that \[\sin\alpha + \cos\alpha = \sqrt{2}\cos\theta\].
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:
\[\sqrt{3} \sin x - \cos x\]
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 sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
The value of \[\sin^2 \frac{5\pi}{12} - \sin^2 \frac{\pi}{12}\]
If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =
tan 3A − tan 2A − tan A =
If \[\cos P = \frac{1}{7}\text{ and }\cos Q = \frac{13}{14}\], where P and Q both are acute angles. Then, the value of P − Q is
The value of \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is
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
If α and β are the solutions of the equation a tan θ + b sec θ = c, then show that tan (α + β) = `(2ac)/(a^2 - c^2)`.
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 sinθ + cosθ = 1, then the value of sin2θ is equal to ______.
If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.
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
