If 8cotθ = 7, then the value of 1+sinθ/cosθ is:

Question

If 8cotθ = 7, then the value of 1+sinθ/cosθ is:

1. A. 1+√113/8
2. B. 7+√113/8
3. C. 1+√113/7
4. D. 8+√113/7 (Correct answer)

Correct Answer

Option D — 8+√113/7

Detailed Solution & Explanation

The correct answer is $\dfrac{8+\sqrt{113}}{7}$.

Key Points

• $\cot\theta=\dfrac{7}{8}\Rightarrow\csc\theta=\sqrt{1+\cot^2\theta}=\dfrac{\sqrt{113}}{8}$, so $\sin\theta=\dfrac{8}{\sqrt{113}}$ and $\cos\theta=\cot\theta\sin\theta=\dfrac{7}{\sqrt{113}}$.
• $\dfrac{1+\sin\theta}{\cos\theta}=\dfrac{1+\frac{8}{\sqrt{113}}}{\frac{7}{\sqrt{113}}}=\dfrac{\sqrt{113}+8}{7}$.

Additional Information

• The identity to carry into the exam is 1 + cot²θ = cosec²θ, which converts a single given ratio into every other one.
• Once cosecθ is known, sinθ is its reciprocal, and cosθ follows from cosθ = cotθ · sinθ without ever finding the angle.
• The companion identities are sin²θ + cos²θ = 1 and 1 + tan²θ = sec²θ.
• A useful shortcut for this expression: (1 + sinθ)/cosθ equals secθ + tanθ, which is the reciprocal of secθ − tanθ.