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Question: Simplify the expression for the trigonometric term: \( \tan {54^o} \) (a) \[\dfrac{{\cos 9 + \sin...

Simplify the expression for the trigonometric term: tan⁡54o\tan {54^o}
(a) cos⁡9+sin⁡9cos⁡9−sin⁡9\dfrac{{\cos 9 + \sin 9}}{{\cos 9 - \sin 9}}
(b) cos⁡9−sin⁡9cos⁡9−sin⁡9\dfrac{{\cos 9 - \sin 9}}{{\cos 9 - \sin 9}}
(c) cos⁡9−sin⁡9cos⁡9+sin⁡9\dfrac{{\cos 9 - \sin 9}}{{\cos 9 + \sin 9}}
(d) cos⁡9+sin⁡9cos⁡9+sin⁡9\dfrac{{\cos 9 + \sin 9}}{{\cos 9 + \sin 9}}

Explanation

Solution

Hint : The given problem revolves around the concepts of trigonometric equations. So, we will use the definition of trigonometric equations especially for compound angles. Here, we have extracted the 5454 in to the (45+9)(45 + 9) where we know the value of (tan⁡45=1)(\tan 45 = 1) and then substituting it in the formula tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan (A + B) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}the desired solution can be obtained.

Complete step-by-step answer :
At the first, extract the number 5454 into the addition so that there will be the trigonometric angle in the demonstration, we get
⇒tan⁡54=tan⁡(45+9)\Rightarrow \tan 54 = \tan (45 + 9)
Now, we can use the formula for the trigonometric ratios for compound angles for tangent ratios, that istan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan (A + B) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}, we can get
Since, substituting the above extracted values in the formula, we get
⇒tan⁡54=tan⁡45+tan⁡91−tan⁡45tan⁡9\Rightarrow \tan 54 = \dfrac{{\tan 45 + \tan 9}}{{1 - \tan 45\tan 9}}
Where, A =45= 45 and B =9= 9 respectively.
\\{ When A=45oA = {45^o} use the formula 1+tan⁡θ1−tan⁡θ\dfrac{{1 + \tan \theta }}{{1 - \tan \theta }} here θ\theta is 99 \\}
⇒tan⁡54=1+tan⁡91−tan⁡9\Rightarrow \tan 54 = \dfrac{{1 + \tan 9}}{{1 - \tan 9}}
According to the trigonometric table of right-angle triangle, we know that tan⁡45=1\tan 45 = 1
⇒tan⁡54=1+sin⁡9cos⁡91−sin⁡9cos⁡9\Rightarrow \tan 54 = \dfrac{{1 + \dfrac{{\sin 9}}{{\cos 9}}}}{{1 - \dfrac{{\sin 9}}{{\cos 9}}}}
Here, we have again extracted the tangent version into the sine and cosine terms for the ease of the solution.
Now, simplifying the equations by adding, multiplying and dividing the terms, we get
⇒tan⁡54=cos⁡9+sin⁡9cos⁡9cos⁡9−sin⁡9cos⁡9\Rightarrow \tan 54 = \dfrac{{\dfrac{{\cos 9 + \sin 9}}{{\cos 9}}}}{{\dfrac{{\cos 9 - \sin 9}}{{\cos 9}}}}
Multiplying and dividing the cosine terms get cancelled that is equals to one,
Therefore , we can write
⇒tan⁡54=cos⁡9+sin⁡9cos⁡9−sin⁡9\Rightarrow \tan 54 = \dfrac{{\cos 9 + \sin 9}}{{\cos 9 - \sin 9}}
∴⇒\therefore \Rightarrow As a result , it can be determined that option (a) is absolutely correct !
So, the correct answer is “Option a”.

Note : One can find the solution by extracting the trigonometric angle so that one of the value/s is the standard angle in terms of the right-angled triangle. Should know the formula for trigonometric ratios for compound angles. Also, we should know all the required values of standard angles say, 0o,30o,45o,60o,90o,180o,270o,360o{0^o},{30^o},{45^o},{60^o},{90^o},{180^o},{270^o},{360^o}respectively for each trigonometric terms such as sin⁡,cos⁡,tan⁡,cot⁡,sec⁡,cos⁡ec\sin ,\cos ,\tan ,\cot ,\sec ,\cos ec , etc. We should take care of the calculations so as to be sure of our final answer.