Question

Find the exact value of each expression. Do not use a calculator.

$\cos {540}^{\circ }-\tan (-{405}^{\circ })$

Short answer

The value of the given expression is 0.

Step-by-step solution

Step 1. Given

Given trigonometric function

$\cos {540}^{\circ }-\tan (-{405}^{\circ })$

Step 2. Concept

We will use standard formula of $\sin \theta$ and $\cos \theta$ at different angles for calculating the final result of given expression.

Step 3. Calculation of cos540∘

We know

$$\begin{gathered} \cos \left(360+\theta \right)=\cos \theta \\ thus, \\ \cos (360+180)=\cos \left(540\right) \\ \cos \left(180\right)=-1 \end{gathered}$$

Step 4. Calculation of tan(-405∘)

We know

$$\begin{gathered} \tan (360+45)=\tan 45 \\ Also \\ \tan (-45)=-\tan \left(45\right) \\ therefore, \\ \tan (-405)=-1 \end{gathered}$$

Step 5. Final calculation

Substituting the values of trigonometric expression we get

$$\begin{gathered} \cos 540-\tan (-405) \\ =-1-(-1) \\ =-1+1 \\ =0 \end{gathered}$$