If p=sin A/1+cos A, then sin A/1-cos A is equal to:

Question

If p=sin A/1+cos A, then sin A/1-cos A is equal to:

1. A. p-1
2. B. 1/p (Correct answer)
3. C. 1/1-p
4. D. 1/p+1

Correct Answer

Option B — 1/p

Detailed Solution & Explanation

The correct answer is $\dfrac{1}{p}$.

Key Points

• Multiply numerator and denominator of $\dfrac{\sin A}{1-\cos A}$ by $(1+\cos A)$: $\dfrac{\sin A(1+\cos A)}{1-\cos^2 A}=\dfrac{\sin A(1+\cos A)}{\sin^2 A}=\dfrac{1+\cos A}{\sin A}$.
• This is the reciprocal of $p=\dfrac{\sin A}{1+\cos A}$, so the value is $\dfrac{1}{p}$.

Additional Information

• The move that solves this is multiplying by the conjugate (1 + cosA), which turns 1 − cos²A into sin²A.
• The result is the pair of identities sinA/(1 + cosA) = (1 − cosA)/sinA and sinA/(1 − cosA) = (1 + cosA)/sinA, which are reciprocals of each other.
• Both equal tan(A/2) and cot(A/2) respectively, which is the half-angle form of the same result.