Question

Prove that $\tan {36^ \circ } + \tan {9^ \circ } + \tan {36^ \circ }\tan {9^ \circ } = 1$

Answer 1

Hint: Here to prove this question we must know property which is mentioned below: -
$\tan (A + B) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$

Complete step-by-step answer:
We know the value of $\tan {45^ \circ } = 1$, so here in this question we will use this value to prove our question.
$\tan {45^ \circ } = 1$
We can write $\tan {45^ \circ }$ as $\tan ({36^ \circ } + {9^ \circ })$ so that property $\tan (A + B) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$ can be applied.
$ \Rightarrow \tan ({36^ \circ } + {9^ \circ }) = \dfrac{{\tan {{36}^ \circ } + \tan {9^ \circ }}}{{1 - \tan {{36}^ \circ }\tan {9^ \circ }}} = 1$
$ \Rightarrow \dfrac{{\tan {{36}^ \circ } + \tan {9^ \circ }}}{{1 - \tan {{36}^ \circ }\tan {9^ \circ }}} = 1$ (Now in next step we will cross multiply)
\[ \Rightarrow \tan {36^ \circ } + \tan {9^ \circ } = 1 - \tan {36^ \circ }\tan {9^ \circ }\]
Rearranging the terms and taking trigonometric functions in one side.
\[\therefore \tan {36^ \circ } + \tan {9^ \circ } + \tan {36^ \circ }\tan {9^ \circ } = 1\]
Hence it is proved that $\tan {36^ \circ } + \tan {9^ \circ } + \tan {36^ \circ }\tan {9^ \circ } = 1$

Additional Information: Trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side length. Most widely used trigonometric functions are sine, cosine and tangent. The angles of sine, cosine and tangent are the primary classification of functions of trigonometry. And the functions which are cotangent, secant and cosecant can be derived from the primary functions. Behaviour of all these functions in four quadrants is as follows: -
First quadrant = All trigonometric functions are positive (sine, cosine, tan, sec, cosec, cot)
Second quadrant=Positive (sine, cosec) Negative (cosine, tan, sec, cot)
Third quadrant= Positive (tan, cot) Negative (sine, cosine, sec, cosec)
Fourth quadrant= Positive (cosine, sec) Negative (sine, tan, cot, cosec)

Note: Students may likely to make one common mistake in this question is that they will try to think the direct value of given trigonometric functions which is a very wrong way to approach because one cannot memorise too many values rather some specific values should be remembered to solve these types of questions. Some of the values are mentioned below: -
$\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}$
$\tan {60^ \circ } = \sqrt 3 $
$\tan {45^ \circ } = 1$
\[\tan {90^ \circ }\]= infinity
$\tan {180^ \circ } = 0$