Prove That: Tan 2 2 X − Tan 2 X 1 − Tan 2 2 X Tan 2 X = Tan 3 X Tan X - Mathematics

Question

Prove that:
$\frac{\tan^{2} 2 x - \tan^{2} x}{1 - \tan^{2} 2 x \tan^{2} x} = \tan 3 x \tan x$

Short Note

Solution

$LHS = \frac{\tan^{2} 2 x - \tan^{2} x}{1 - \tan^{2} 2 x \tan^{2} x}$
$= \frac{(\tan 2 x + \tan x) (\tan 2 x - \tan x)}{1 - \tan^{2} 2 x \tan^{2} x} {Using A^{2} - B^{2} = (A + B) (A - B)}$
$\tan 3 x = \tan (2 x + x) and \tan x = \tan (2 x - x) .$
$= \frac{\tan 3 x (1 - \tan 2 x \tan x) \times \tan x (1 + \tan 2 x \tan x)}{1 - \tan^{2} 2 x \tan^{2} x} [∵ \tan 2 x + \tan x = \tan 3 x (1 - \tan 2 x \tanx) \tan 2 x - \tanx = \tanx (1 + \tan 2 x \tanx)]$
$= \frac{\tan 3 x \tan x (1 - \tan^{2} 2 x \tan^{2} x)}{1 - \tan^{2} 2 x \tan^{2} x}$
$= \tan 3 x \tan x$
= RHS
Hence proved .

shaalaa.com

Trigonometric Functions of Sum and Difference of Two Angles

Is there an error in this question or solution?

Q 18 Q 17.4 Q 19

Chapter 7: Values of Trigonometric function at sum or difference of angles - Exercise 7.1 [Page 20]

APPEARS IN

RD Sharma Mathematics [English] Class 11

Chapter 7 Values of Trigonometric function at sum or difference of angles
Exercise 7.1 | Q 18 | Page 20

Video TutorialsVIEW ALL [1]

view
Video Tutorials For All Subjects
Trigonometric Functions of Sum and Difference of Two Angles

video tutorial00:06:30

RELATED QUESTIONS

Prove the following: $\cos (\frac{π}{4} \times x) \cos (\frac{π}{4} - y) - \sin (\frac{π}{4} - x) \sin (\frac{π}{4} - y) = \sin (x + y)$

Prove the following:

sin 2x + 2sin 4x + sin 6x = 4cos² x sin 4x

Prove the following:

cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)

Prove the following:

$\frac{\sin x - \sin y}{\cos x + \cos y} = \tan \frac{x - y}{2}$

If $\sin A = \frac{4}{5}$ and $\cos B = \frac{5}{13}$ , where 0 < A, $B < \frac{π}{2}$ , find the value of the following:

sin (A + B)

If $\sin A = \frac{4}{5}$ and $\cos B = \frac{5}{13}$ , where 0 < A, $B < \frac{π}{2}$ , find the value of the following:
cos (A + B)

If $\sin A = \frac{4}{5}$ and $\cos B = \frac{5}{13}$ , where 0 < A, $B < \frac{π}{2}$ , find the value of the following:
sin (A − B)

If $\cos A = - \frac{24}{25} and \cos B = \frac{3}{5}$ , where π < A < $\frac{3 π}{2} and \frac{3 π}{2}$ < B < 2π, find the following:
sin (A + B)

If $\cos A = - \frac{24}{25} and \cos B = \frac{3}{5}$ , where π < A < $\frac{3 π}{2} and \frac{3 π}{2}$ < B < 2π, find the following:
cos (A + B)

Evaluate the following:
sin 78° cos 18° − cos 78° sin 18°

Evaluate the following:
sin 36° cos 9° + cos 36° sin 9°

If $\tan A = \frac{m}{m - 1} and \tan B = \frac{1}{2 m - 1}$ , then prove that $A - B = \frac{π}{4}$ .

Prove that:
sin² B = sin² A + sin² (A − B) − 2 sin A cos B sin (A − B)

Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1

If sin (α + β) = 1 and sin (α − β) $= \frac{1}{2}$ , where 0 ≤ α, $β \leq \frac{π}{2}$ , 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

\cos (α + β) = \frac{b^{2} - a^{2}}{b^{2} + a^{2}}

Prove that:

\frac{1}{\sin (x - a) \cos (x - b)} = \frac{\cot (x - a) + \tan (x - b)}{\cos (a - b)}

Reduce each of the following expressions to the sine and cosine of a single expression:

cos x − sin 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 $(x + \frac{π}{3}) + 3$ lies.

If A + B = C, then write the value of tan A tan B tan C.

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 $\cos P = \frac{1}{7} and \cos Q = \frac{13}{14}$ , where P and Q both are acute angles. Then, the value of P − Q is

If tan (π/4 + x) + tan (π/4 − x) = a, then tan² (π/4 + x) + tan² (π/4 − x) =

The maximum value of $\sin^{2} (\frac{2 π}{3} + x) + \sin^{2} (\frac{2 π}{3} - x)$ is

Express the following as the sum or difference of sines and cosines:
2 cos 7x cos 3x

If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.

The value of sin(45° + θ) - cos(45° - θ) is ______.

If tanA = $\frac{1}{2}$ , tanB = $\frac{1}{3}$ , then tan(2A + B) is equal to ______.

If sinx + cosx = a, then |sinx – cosx| = ______.

Given x > 0, the values of f(x) = $- 3 \cos \sqrt{3 + x + x^{2}}$ lie in the interval ______.

State whether the statement is True or False? Also give justification.

If tanθ + tan2θ + $\sqrt{3}$ tanθ tan2θ = $\sqrt{3}$ , then θ = $\frac{n π}{3} + \frac{π}{9}$

Prove That: Tan 2 2 X − Tan 2 X 1 − Tan 2 2 X Tan 2 X = Tan 3 X Tan X - Mathematics

Advertisements

Advertisements

Question

Solution

APPEARS IN

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Select a course

Prove That: Tan 2 2 X − Tan 2 X 1 − Tan 2 2 X Tan 2 X = Tan 3 X Tan X - Mathematics

Advertisements

Advertisements

Question

SolutionShow Solution

APPEARS IN

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Select a course

Solution