Question

If $\angle x = {30^ \circ }$, then
$\tan 2x = \dfrac{{2\tan x}}{{1 - {{\tan }^2}x}}$
If true enter $1$ else $0$

Answer 1

To solve this question first solve the right-hand side of the equation by putting the value $\angle x = {30^ \circ }$ then simplify that equation to come to the shortest answer that is possible. Then put the value on the left-hand side and find the value after putting that value. If both the sides are equal then enter 1 and if they are not equal. Then enter 0.

Complete answer:
Given,
Angle $x$ is given $\angle x = {30^ \circ }$and the expression is also given $\tan 2x = \dfrac{{2\tan x}}{{1 - {{\tan }^2}x}}$
To, solve this question first solve right hand side of the equation by putting the value $\angle x = {30^ \circ }$
The right hand side of the equation is
$RHS = \dfrac{{2\tan x}}{{1 - {{\tan }^2}x}}$
On putting the value of $x$
$\Rightarrow \dfrac{{2\tan \left( {30} \right)}}{{1 - {{\tan }^2}30}}$
We know that value of $\tan 30 = \dfrac{1}{{\sqrt 3 }}$ on putting this value
$\Rightarrow \dfrac{{2\dfrac{1}{{\sqrt 3 }}}}{{1 - {{\left( {\dfrac{1}{{\sqrt 3 }}} \right)}^2}}}$
On further solving
$\Rightarrow \dfrac{{\dfrac{2}{{\sqrt 3 }}}}{{1 - \dfrac{1}{3}}}$
On taking LCM in denominator
$\Rightarrow \dfrac{{\dfrac{2}{{\sqrt 3 }}}}{{\dfrac{{3 - 1}}{3}}}$
On further solving
$\Rightarrow \dfrac{2}{{\sqrt 3 }} \times \dfrac{3}{2}$
$\Rightarrow \sqrt 3$
The value of left hand side of the equation:
$RHS = \sqrt 3$ ……(i)
On putting the value of $x$ in left hand side
$LHS = \tan 2x$
$\Rightarrow \tan (2 \times 30)$
$\Rightarrow \tan (60)$
We know that $\tan ({60^ \circ }) = \sqrt 3$
On putting this value we get value of left hand side
$LHS = \sqrt 3$ ……(ii)
From equation (i) and (ii) the value of LHS and RHS are equal to we enter 1
Final answer:
From equations (i) and (ii) we get that the left-hand side and right-hand side both are equal.

Note:
To solve these types of questions we have to directly put the value in the given expression and find their values on the left-hand side and right-hand side. If both the sides are equal then the condition is true and if they are not equal then that condition is not true.