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

Hint: 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.