Question

Find the value of \[\tan 100^\circ + \tan 125^\circ + \tan 100^\circ \cdot \tan 125^\circ \]
A) $0$
B) \[\dfrac{1}{2}\]
C) \[ - 1\]
D) $1$

Answer 1

Hint: Here we will use the trigonometric formula of the tan function of the sum of two angles. We will substitute the measure of angles in the formula. Then we will solve the equation to get the required value.

Complete step by step solution:
Given equation is \[\tan 100^\circ + \tan 125^\circ + \tan 100^\circ \cdot \tan 125^\circ \].
We will use a trigonometric property of the tan function to get the value of the above equation. We know that\[\tan \left( {a + b} \right) = \dfrac{{\tan a + \tan b}}{{1 - \tan a \cdot \tan b}}\].
Substituting \[a\] as \[100^\circ \] and \[b\] as \[125^\circ \] in the formula \[\tan \left( {a + b} \right) = \dfrac{{\tan a + \tan b}}{{1 - \tan a \cdot \tan b}}\], we get
\[\tan \left( {100^\circ + 125^\circ } \right) = \dfrac{{\tan 100^\circ + \tan 125^\circ }}{{1 - \tan 100^\circ \cdot \tan 125^\circ }}\]
Adding the angles on LHS, we get
\[ \Rightarrow \tan \left( {225^\circ } \right) = \dfrac{{\tan 100^\circ + \tan 125^\circ }}{{1 - \tan 100^\circ \cdot \tan 125^\circ }}\]
We can write \[\tan \left( {225^\circ } \right)\] as \[\tan \left( {180^\circ + 45^\circ } \right)\]. Therefore, we get
\[ \Rightarrow \tan \left( {180^\circ + 45^\circ } \right) = \dfrac{{\tan 100^\circ + \tan 125^\circ }}{{1 - \tan 100^\circ \cdot \tan 125^\circ }}\]
We know that the tan function in the third quadrant is positive i.e. \[\tan \left( {180^\circ + 45^\circ } \right) = \tan 45^\circ \]. Therefore,
\[ \Rightarrow \tan \left( {45^\circ } \right) = \dfrac{{\tan 100^\circ + \tan 125^\circ }}{{1 - \tan 100^\circ \cdot \tan 125^\circ }}\]
Now we know that \[\tan \left( {45^\circ } \right) = 1\], therefore we get
\[ \Rightarrow 1 = \dfrac{{\tan 100^\circ + \tan 125^\circ }}{{1 - \tan 100^\circ \cdot \tan 125^\circ }}\]
On cross multiplication, we get
\[ \Rightarrow 1 - \tan 100^\circ \cdot \tan 125^\circ = \tan 100^\circ + \tan 125^\circ \]
Now taking all the tangent function on RHS, we get
\[ \Rightarrow 1 = \tan 100^\circ + \tan 125^\circ + \tan 100^\circ \cdot \tan 125^\circ \]
\[ \Rightarrow \tan 100^\circ + \tan 125^\circ + \tan 100^\circ \cdot \tan 125^\circ = 1\]

Hence the value of \[\tan 100^\circ + \tan 125^\circ + \tan 100^\circ \cdot \tan 125^\circ \] is equal to $1$. Therefore, option (D) is the correct option.

Note: Here instead of using the values from the trigonometric table, we used the formula of the tangent function of sum of two angles. Using the formula it becomes convenient to transform it into a given expression. We should also know the different properties of the trigonometric function and also in which quadrant which function is positive or negative as in the first quadrant all the functions i.e. sin, cos, tan, cot, sec, cosec is positive. In the second quadrant, only the sin and cosec function are positive and all the other functions are negative. In the third quadrant, only tan and cot function is positive and in the fourth quadrant, only cos and sec function is positive.