Question

costheta+sintheta/costheta-sintheta

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{\dfrac{cos\,\theta+sin\,\theta}{cos\,\theta-sin\,\theta}}$

$\underline{\textbf{To simplify:}}$

$\mathsf{\dfrac{cos\,\theta+sin\,\theta}{cos\,\theta-sin\,\theta}}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{\dfrac{cos\,\theta+sin\,\theta}{cos\,\theta-sin\,\theta}}$

$\textsf{This can be written as}$

$\mathsf{=\dfrac{cos\,\theta\left(1+\dfrac{sin\,\theta}{cos\,\theta}\right)}{cos\theta\left(1--\dfrac{sin\,\theta}{cos\,\theta}}\right)}$

$\mathsf{=\dfrac{1+tan\,\theta}{1--tan\,\theta}}$

$\mathsf{=\dfrac{tan\frac{\pi}{4}+tan\,\theta}{1--tan\frac{\pi}{4}\;tan\,\theta}}$

$\mathsf{using,}$

$\boxed{\mathsf{tan(A+B)=\dfrac{tanA+tanB}{1-tanA\,tanB}}}$

$\mathsf{=tan\left(\dfrac{\pi}{4}+\theta\right)}$

$\implies\boxed{\mathsf{\dfrac{cos\,\theta+sin\,\theta}{cos\,\theta-sin\,\theta}=tan\left(\dfrac{\pi}{4}+\theta\right)}}$

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