Question

The set of values of x for which the expression \[\dfrac{{\tan 3x - \tan 2x}}{{1 + \tan 3x\tan 2x}} = 1\] is
A. \[\ell \] \[\begin{array}{l}\ell \\\dfrac{\pi }{4}\\\{ n\pi + \dfrac{\pi }{4}:n = 1,2,3.......\} \\\{ 2n\pi + \dfrac{\pi }{4}:n = 1,2,3.......\} \end{array}\]
B. \[\dfrac{\pi }{4}\]
C. \[\{ n\pi + \dfrac{\pi }{4}:n = 1,2,3.......\} \]
D. \[\{ 2n\pi + \dfrac{\pi }{4}:n = 1,2,3.....\} \]

Answer 1

Hint: Before we proceed into the problem, it is important to know the definitions of trigonometry. The area of mathematics known as trigonometry deals with particular angles' mathematical functions and how to use them six different trigonometric functions can be applied to a common angle. Sine, cosine, tangent, cotangent, secant, and cosecant are their names, respectively, as well as their acronyms \[\left( {cosec} \right)\]. Trigonometry is used to establish directions such as the north, south, east, and west; it indicates the direction to point the compass in order to travel in a straight line. It helps to locate a certain location during navigation. The shore's separation from a certain place in the sea can likewise be determined using this method.

Complete step by step solution:\[\dfrac{{\tan 3x - \tan 2x}}{{1 + \tan 3x\tan 2x}} = 1\]
⟹\[\tan (3x - 2x) = 1\]
⟹\[\tan x = 1\]
Provided \[\tan 2x\tan 3x = - 1\]
\[n\pi + \dfrac{\pi }{4}\]
\[x\]If solve the above equation, we get \[x\] to be
But that will make \[\tan 2x\] to be infinite, and therefore, this value cannot be accepted
So, there is no such \[x\] satisfying the above equality.

Option ‘A’ is correct

Note: All trigonometric formulas require one of the six fundamental trigonometric ratios. These ratios, which are also referred to as trigonometric functions, typically employ all trigonometric formulas. The sine, cosine, secant, cosecant, tangent, and cotangent are the six fundamental trigonometric functions. The need to calculate angles and distances in disciplines like astronomy, mapmaking, surveying, and artillery range finding led to the development of trigonometry. Plane trigonometry deals with issues involving angles and lengths in a single plane. Spherical trigonometry takes into account applications to similar issues in more than one plane of three-dimensional space.