Question

Find the value of \[\dfrac{{\left[ {1 - {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}{{\left[ {1 + {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}\].
A. \[\sin 2A\]
B. \[\cos 2A\]
C. \[\tan 2A\]
D. \[\cot 2A\]

Answer 1

Hint: In this question, we need to find the value of \[\dfrac{{\left[ {1 - {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}{{\left[ {1 + {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}\]. For this, we have to convert \[\tan \] in terms of \[\sin \] and \[\cos \] first. After that we will get the final result by simplifying it using trigonometric identities.

Formula used: We will use the following trigonometric identities for solving this example.
1. \[\tan A = \dfrac{{sinA}}{{cosA}}\]
2. \[{\sin ^2}A + {\cos ^2}A = 1\]
3. \[{\cos ^2}A - {\sin ^2}A = \cos 2A\]

Complete step-by-step answer:
We know that the given expression is \[\dfrac{{\left[ {1 - {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}{{\left[ {1 + {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}\]
Now, we will simplify the above expression.
First, we will convert tangent terms into sin and cos terms.
\[
  \dfrac{{\left[ {1 - {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}{{\left[ {1 + {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}} = \dfrac{{\left[ {1 - \dfrac{{{{\sin }^2}\left( {{{45}^ \circ } - A} \right)}}{{{{\cos }^2}\left( {{{45}^ \circ } - A} \right)}}} \right]}}{{\left[ {1 + \dfrac{{{{\sin }^2}\left( {{{45}^ \circ } - A} \right)}}{{{{\cos }^2}\left( {{{45}^ \circ } - A} \right)}}} \right]}} \\
   = \dfrac{{\left[ {\dfrac{{{{\cos }^2}\left( {{{45}^ \circ } - A} \right) - {{\sin }^2}\left( {{{45}^ \circ } - A} \right)}}{{{{\cos }^2}\left( {{{45}^ \circ } - A} \right)}}} \right]}}{{\left[ {\dfrac{{{{\cos }^2}\left( {{{45}^ \circ } - A} \right) + {{\sin }^2}\left( {{{45}^ \circ } - A} \right)}}{{{{\cos }^2}\left( {{{45}^ \circ } - A} \right)}}} \right]}} \\
 \]
But we know that \[{\sin ^2}A + {\cos ^2}A = 1\]and \[{\cos ^2}A - {\sin ^2}A = \cos 2A\]
By applying these identities, we get
\[
  \dfrac{{\left[ {1 - {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}{{\left[ {1 + {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}} = \dfrac{{\left[ {\dfrac{{\cos 2\left( {{{45}^ \circ } - A} \right)}}{{{{\cos }^2}\left( {{{45}^ \circ } - A} \right)}}} \right]}}{{\left[ {\dfrac{1}{{{{\cos }^2}\left( {{{45}^ \circ } - A} \right)}}} \right]}} \\
   = \left[ {\dfrac{{\cos 2\left( {{{45}^ \circ } - A} \right)}}{{{{\cos }^2}\left( {{{45}^ \circ } - A} \right)}}} \right] \times \dfrac{{{{\cos }^2}\left( {{{45}^ \circ } - A} \right)}}{1} \\
   = \cos \left( {{{90}^ \circ } - 2A} \right) \\
 \]
Bu \[\cos \left( {{{90}^ \circ } - 2A} \right) = \sin 2A\]
By applying this identity, we get
\[\dfrac{{\left[ {1 - {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}{{\left[ {1 + {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}} = \sin 2A\]
Hence, the value of the expression \[\dfrac{{\left[ {1 - {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}{{\left[ {1 + {{\tan }^2}\left( {{{45}^ \circ } - A} \right)} \right]}}\]is \[\sin 2A\].
Therefore, the correct option is (A).

Additional Information: Trigonometric Identities are equalities that utilize trigonometry functions and remain true for all values of variables specified in the equation. Trigonometric Identities help a lot when trigonometric functions can be used in an expression or equation. Trigonometric laws hold for all values of variables on both sides of an equation. There are several unique trigonometric identities relating the side length and angle of a triangle. The trigonometric identities are only valid for right-angle triangles. There are several trigonometric identities available to simplify the crucial part of problems.

Note: Many students make mistakes in the simplification part and applying proper trigonometric identity. Complex trigonometric problems can be addressed rapidly using these trigonometric identities or formulae.