Question

The value of $\tan {45^ \circ } \times \tan {135^ \circ }$ is equal to
A. -1
B. 0
C. 1
D. None of above

Answer 1

Hint: For solving this type of questions, we will use trigonometric identities. Here we will find out the value of $\tan {45^ \circ }$ and $\tan {135^ \circ }$.We will find out the value of $\tan {135^ \circ }$ by using the trigonometric identity $\tan (90 + \theta )$ and solve accordingly.

Complete step-by-step answer:
The tangent function is a periodic function which is very important in trigonometry. The simplest way to understand the tangent function is to use the unit circle. For a given angle measure $\theta$ draw a unit circle on the coordinate plane and draw the angle centred at the origin, with one side as the positive x -axis. The x -coordinate of the point where the other side of the angle intersects the circle is cos($\theta$) and the y -coordinate is sin($\theta$).
$\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
Some standard values of $\tan \theta$ (where $\theta$ is measured in degrees) are as follows:
The value of tan ${0^ \circ }$ = 0
The value of tan ${30^ \circ }$ = $\dfrac{{\sqrt 3 }}{2}$
The value of tan ${45^ \circ }$ = 1
The value of tan ${60^ \circ }$ = $\sqrt 3$
The value of tan ${90^ \circ }$ = Not defined ($\infty$)
We know that value of tan ${45^ \circ }$ = 1,
We know, $\tan (90 + \theta ) = - \cot \theta$.
Therefore, $\tan ({135^ \circ }) = \tan {(90 + 45)^ \circ } = - \cot ({45^ \circ })$.
As we know that: $\cot \theta = \dfrac{1}{{\tan \theta }}$
Therefore, $- \cot {45^ \circ } = \dfrac{1}{{\tan {{45}^ \circ }}}$.
Substituting the value of tan ${45^ \circ }$, we get, $- \cot {45^ \circ } = - 1$
Therefore,
$\tan {45^ \circ } \times \tan {135^ \circ } = 1 \times - 1 = - 1$
Hence, the correct answer is option (A) -1.

Note: The basic problem faced in this type of questions is how to use trigonometric identities. For solving this question we should know the basic trigonometric identities like $\tan (90 + \theta )$and $\tan (180 - \theta )$.