For an acute angle \(\theta\), the expression \(\displaystyle\frac{\cos\theta}{1-\sin\theta}+\frac{\cos\theta}{1+\sin\theta}\) simplifies to which multiple of a single trigonometric ratio?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(2\sec\theta\)
Take the common denominator \((1-\sin\theta)(1+\sin\theta)=1-\sin^2\theta\), which by \(\sin^2\theta+\cos^2\theta=1\) is \(\cos^2\theta\). In the given range \(\sin\theta\ne 1\) and \(\cos\theta\ne 0\), so nothing is divided by zero.
Numerator: \(\cos\theta(1+\sin\theta)+\cos\theta(1-\sin\theta)=2\cos\theta\)
Denominator: \(1-\sin^2\theta=\cos^2\theta\)
Expression \(=\displaystyle\frac{2\cos\theta}{\cos^2\theta}=\frac{2}{\cos\theta}=2\sec\theta\)