Question

For which value of acute angle
Cos theta/1-sin theta+cos theta/1+sin theta=4 is true?

Answer 1

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

$\mathsf{\dfrac{cos\,\theta}{1-sin\,\theta}+\dfrac{cos\,\theta}{1+sin\,\theta}=4}$

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

$\textsf{The value of angle}\;\mathsf{\theta}\;\textsf{satisfying the given equation}$

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

$\underline{\textbf{Consider:}}$

$\mathsf{\dfrac{cos\,\theta}{1-sin\,\theta}+\dfrac{cos\,\theta}{1+sin\,\theta}=4}$

$\implies\mathsf{cos\,\theta\left(\dfrac{1}{1-sin\,\theta}+\dfrac{1}{1+sin\,\theta}\right)}=4}$

$\implies\mathsf{cos\,\theta\left(\dfrac{1+sin\,\theta+1+sin\,\theta}{(1-sin\,\theta)(1+sin\,\theta)}\right)=4}$

$\implies\mathsf{cos\,\theta\left(\dfrac{2}{1-sin^2\,\theta}\right)=4}$

$\implies\mathsf{cos\,\theta\left(\dfrac{2}{cos^2\,\theta}\right)=4}$

$\implies\mathsf{\dfrac{2}{cos\,\theta}=4}$

$\implies\mathsf{\dfrac{2}{4}=cos\,\theta}$

$\implies\mathsf{cos\,\theta=\dfrac{1}{2}}$

$\implies\boxed{\mathsf{\theta=60^\circ}}$