Find the value of $\cos 135{}^\circ $.

Hint: We know that the value of $\cos \theta $ is positive in the first and fourth quadrant and negative in the second quadrant and third quadrant. We also have the identity that can be used to simplify and find the value. So, we will use the identity given by $\cos \left( 90{}^\circ +\theta \right)=-\sin \theta $ to simplify and find the value of $\cos 135{}^\circ $.

Complete step-by-step answer:
It is given in the question that we have to find the value of $\cos 135{}^\circ $. We know that the value of $\cos \theta $ decreases with the increase in the value of $\theta $. We also know that the value of $\cos \theta $ is positive in the first quadrant and negative in the second quadrant. We also know that, $\cos \left( 90{}^\circ +\theta \right)=-\sin \theta $. So, we will use this formula to find the value of $\cos 135{}^\circ $. We can write $\cos 135{}^\circ $ as,
 $\cos \left( 90{}^\circ +45{}^\circ \right)=-\sin 45{}^\circ $
Now, we know that the value of $\sin 45{}^\circ =\dfrac{1}{\sqrt{2}}$. So, $-\sin 45{}^\circ =-\dfrac{1}{\sqrt{2}}$. Therefore, we get the above equality as,
$\cos \left( 90{}^\circ +45{}^\circ \right)=-\dfrac{1}{\sqrt{2}}$ or we can say that, $\cos 135{}^\circ =-\dfrac{1}{\sqrt{2}}$.

Therefore, the value of $\cos 135{}^\circ =-\dfrac{1}{\sqrt{2}}$.

Note: The students generally skip the - or negative sign with $\dfrac{1}{\sqrt{2}}$, but we know that it is not correct as the value of $\cos \theta $ is negative in the second quadrant. Many students also tend to write the formula incorrectly as, $\cos \left( 90{}^\circ +\theta \right)=\sin \theta $. Thus, it is advisable that the students should remember all the basic trigonometric formulas to solve such questions.