Prove that: $\dfrac{\cos A}{1+\sin A}+\dfrac{1+\sin A}{\cos A}=2\sec A$.
Step-by-Step Solution
Key Concept: Combine the two fractions on the LHS over a common denominator, then simplify the numerator using the Pythagorean identity.
LHS $=\dfrac{\cos A}{1+\sin A}+\dfrac{1+\sin A}{\cos A}$. Taking the common denominator $\cos A(1+\sin A)$: [1.0 Mark]
$=\dfrac{\cos^2A+(1+\sin A)^2}{\cos A(1+\sin A)}$. [1.0 Mark]
Expand the numerator: $\cos^2A+1+2\sin A+\sin^2A=(\sin^2A+\cos^2A)+1+2\sin A=1+1+2\sin A=2+2\sin A$. [1.5 Marks]
$=\dfrac{2(1+\sin A)}{\cos A(1+\sin A)}=\dfrac{2}{\cos A}$. [1.0 Mark]
$=2\sec A$. Hence proved. [0.5 Mark]
Correct Answer: