Question

The value of \[8184\left[ {\sin 12^\circ \sin 48^\circ \sin 54^\circ } \right] + 181\left[ {\tan 203^\circ + \tan 22^\circ + \tan 203^\circ \tan 22^\circ } \right]\] is equal to.

Answer 1

Hint: Start with the first bracket, apply the property \[2\sin A\sin B = \cos \left( {A - B} \right)-\cos \left( {A + B} \right)\] in the first two terms. After this, apply the property \[\tan A + \tan B = \tan \left( {A + B} \right)\left( {1 - \tan A\tan B} \right)\] in the first two terms of the second bracket. Simplify by putting the trigonometric values.

Complete step-by-step answer:
Consider the given expression,
\[8184\left[ {\sin 12^\circ \sin 48^\circ \sin 54^\circ } \right] + 181\left[ {\tan 203^\circ + \tan 22^\circ + \tan 203^\circ \tan 22^\circ } \right]\]
We will first simplify the first bracket by using the trigonometric identity \[2\sin A\sin B = \cos \left( {A - B} \right)\cos \left( {A + B} \right)\] on the first two terms.
Thus, we get,
\[
   \Rightarrow 2 \times 4092\left[ {\sin 12^\circ \sin 48^\circ \sin 54^\circ } \right] + 181\left[ {\tan 203^\circ + \tan 22^\circ + \tan 203^\circ \tan 22^\circ } \right] \\
   \Rightarrow 4092\left[ {\left( {2\sin 48^\circ \sin 12^\circ } \right)\sin 54^\circ } \right] + 181\left[ {\tan 203^\circ + \tan 22^\circ + \tan 203^\circ \tan 22^\circ } \right] \\
   \Rightarrow 4092\left[ {\cos \left( {48^\circ - 12^\circ } \right) - \cos \left( {48^\circ + 12^\circ } \right)} \right]\sin 54^\circ + 181\left[ {\tan 203^\circ + \tan 22^\circ + \tan 203^\circ \tan 22^\circ } \right] \\
   \Rightarrow 4092\left[ {\cos \left( {36^\circ } \right) - \cos \left( {60^\circ } \right)} \right]\sin 54^\circ + 181\left[ {\tan 203^\circ + \tan 22^\circ + \tan 203^\circ \tan 22^\circ } \right] \\
 \]
Now, we know that \[\sin \left( {90^\circ - 54^\circ } \right) = \cos 36^\circ \]
We will put this value in the above obtained expression,
Thus, we get,
\[ \Rightarrow 4092\left[ {\cos \left( {36^\circ } \right) - \cos \left( {60^\circ } \right)} \right]\cos 36^\circ + 181\left[ {\tan 203^\circ + \tan 22^\circ + \tan 203^\circ \tan 22^\circ } \right]\]
Next, we will simplify the second bracket by using the trigonometric identity \[\tan A + \tan B = \tan \left( {A + B} \right)\left( {1 - \tan A\tan B} \right)\] on the first two terms,
Thus, we have,
\[
   \Rightarrow 4092\left[ {\cos \left( {36^\circ } \right) - \cos \left( {60^\circ } \right)} \right]\cos 36^\circ + 181\left[ {\tan \left( {203^\circ + 22^\circ } \right)\left( {1 - \tan 203^\circ \tan 22^\circ } \right) + \tan 203^\circ \tan 22^\circ } \right] \\
   \Rightarrow 4092\left[ {\cos \left( {36^\circ } \right) - \cos \left( {60^\circ } \right)} \right]\cos 36^\circ + 181\left[ {\tan \left( {225^\circ } \right)\left( {1 - \tan 203^\circ \tan 22^\circ } \right) + \tan 203^\circ \tan 22^\circ } \right] \\
 \]
Since, we know that \[\tan \left( {225^\circ } \right) = 1\] and \[\cos \left( {60^\circ } \right) = \dfrac{1}{2}\]
Hence, put the values in the derived form,
Thus, we get,
\[
   \Rightarrow 4092\left[ {\cos \left( {36^\circ } \right) - \dfrac{1}{2}} \right]\cos 36^\circ + 181\left[ {1\left( {1 - \tan 203^\circ \tan 22^\circ } \right) + \tan 203^\circ \tan 22^\circ } \right] \\
   \Rightarrow 4092\left[ {\cos \left( {36^\circ } \right) - \dfrac{1}{2}} \right]\cos 36^\circ + 181\left[ {1 - \tan 203^\circ \tan 22^\circ + \tan 203^\circ \tan 22^\circ } \right] \\
   \Rightarrow 4092\left[ {\cos \left( {36^\circ } \right) - \dfrac{1}{2}} \right]\cos 36^\circ + 181\left[ 1 \right] \\
 \]
Since, we know that \[\cos \left( {36^\circ } \right) = \dfrac{{\sqrt 5 + 1}}{4}\],
Therefore, substitute the value in the obtained above expression,
We get,
\[
   \Rightarrow 4092\left[ {\dfrac{{\sqrt 5 + 1}}{4} - \dfrac{1}{2}} \right]\left( {\dfrac{{\sqrt 5 + 1}}{4}} \right) + 181 \\
   \Rightarrow 4092\left[ {{{\left( {\dfrac{{\sqrt 5 + 1}}{4}} \right)}^2} - \dfrac{1}{2}\left( {\dfrac{{\sqrt 5 + 1}}{4}} \right)} \right] + 181 \\
   \Rightarrow 4092\left[ {\dfrac{1}{{16}}\left( {1 + 5 + 2\sqrt 5 } \right) - \dfrac{1}{8}\left( {1 + \sqrt 5 } \right)} \right] + 181 \\
   \Rightarrow \dfrac{{4092}}{{16}}\left[ {\left( {6 + 2\sqrt 5 } \right) - 2 - 2\sqrt 5 } \right] + 181 \\
   \Rightarrow 1023 + 181 \\
   \Rightarrow 1204 \\
 \]
Hence, the value of the given expression is equal to 1204.

Note: We can find the value of \[\cos \left( {36^\circ } \right)\] by deriving its value in rough or we can remember the value also. The trigonometric identities \[2\sin A\sin B = \cos \left( {A - B} \right)\cos \left( {A + B} \right)\] and \[\tan A + \tan B = \tan \left( {A + B} \right)\left( {1 - \tan A\tan B} \right)\] should be used to simplify the given expression.