Question

If $y=\left(1+\tan A\right)\left(1-\tan B\right)$, where $A-B=\frac{\mathrm{\pi }}{4}$, then ${\left(y+1\right)}^{y+1}$ is equal to

• A. $9$
• B. $4$
• C. $27$
• D. $81$

Answer 1

The correct option is C

$27$

Explanation for the correct option.

Step 1. Find the value of $y$.

It is given that $A-B=\frac{\mathrm{\pi }}{4}$. So we take tan on both sides:

$\begin{array}{rcl}\tan \left(A-B\right)& =& \tan \left(\frac{\mathrm{\pi }}{4}\right)\\ & \Rightarrow & \frac{\tan A-\tan B}{1+\tan A\tan B}=1\\ & \Rightarrow & \tan A-\tan B=1+\tan A\tan B...\left(1\right)\end{array}$

Now expand the expression $y=\left(1+\tan A\right)\left(1-\tan B\right)$

$$\begin{gathered} y=\left(1+\tan A\right)\left(1-\tan B\right) \\ \Rightarrow y=1-\tan B+\tan A-\tan A\tan B \\ \Rightarrow y=(\tan A-\tan B)+(1-\tan A\tan B) \end{gathered}$$

Using the equation $\left(1\right)$ substitute $1+\tan A\tan B$ for $\tan A-\tan B$.

$\begin{array}{rcl}y& =& 1+\tan A\tan B+1-\tan A\tan B\\ & =& 1+1\\ & =& 2\end{array}$

Step 2. Find the value of the given expression.

In the expression ${\left(y+1\right)}^{y+1}$ substitute $2$ for $y$ and simplify.

$\begin{array}{rcl}{\left(y+1\right)}^{y+1}& =& {\left(2+1\right)}^{2+1}\\ & =& 3^3\\ & =& 27\end{array}$

So, the value of the given expression is $27$.

Hence, the correct option is C.