Question

$\frac{\sin (B+A)+\cos (B-A)}{\sin (B-A)+\cos (B+A)}=$

• A. $\frac{(\cos B+\sin B)}{(\cos B–\sin B)}$
• B. $\frac{(\cos A+\sin A)}{(\cos A–\sin A)}$
• C. $\frac{(\cos A–\sin A)}{(\cos A+\sin A)}$
• D. None of these

Answer 1

The correct option is B

$\frac{(\cos A+\sin A)}{(\cos A–\sin A)}$

Explanation for the correct option:

Simplifying the equations using trigonometric identities:

$\frac{\sin (B+A)+\cos (B-A)}{\sin (B-A)+\cos (B+A)}$

Using the formulas,

$$\begin{gathered} \mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{y}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{y}\mathbf{}\mathbf{} \\ \mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{x}\mathbf{-}\mathit{y}\mathbf{)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{y}\mathbf{-}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{y}\mathbf{} \\ \mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{y}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{y}\mathbf{} \\ \mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathit{x}\mathbf{-}\mathit{y}\mathbf{)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{y}\mathbf{-}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{y} \end{gathered}$$

$$\begin{gathered} =\frac{\sin B\cos A+\cos B\sin A+\cos B\cos A+\sin B\sin A}{\sin B\cos A–\cos B\sin A+\cos B\cos A–\sin B\sin A}\mathbf{}\left[sin(x+y)=cosxsiny+sinxcosy;sin(x-y)=cosxsiny-sinxcosy; \\ cos(x+y)=cosxcosy+sinxsiny;cos(x-y)=cosxcosy-sinxsiny\right] \\ =\frac{\cos A\left(\sin B+\cos B\right)+\sin A\left(\cos B+\sin B\right)}{\cos A(\sin B+\cos B)–\sin A(\cos B+\sin B)} \\ =\frac{(\sin B+\cos B)(\cos A+\sin A)}{(\sin B+\cos B)(\cos A–\sin A)} \\ =\frac{\cos A+\sin A}{\cos A-\sin A} \end{gathered}$$

Therefore, the correct answer is option (B).