Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>Find the value of <strong>m</strong> where <strong>m</strong> = tan²(sec⁻¹ 2) + cot²(cosec⁻¹ 3), then find m² + m + 10</p>
Step-by-Step Solution
Key Concept: Use the identities tan²(θ) = sec²(θ) - 1 and cot²(φ) = cosec²(φ) - 1 to express the sum in terms of secant and cosecant values.
**Step 1:** Let $\theta = \sec^{-1} 2$ and $\phi = \csc^{-1} 3$.
From these definitions, we have $\sec \theta = 2$ and $\csc \phi = 3$.
Squaring these values, we obtain $\sec^2 \theta = 4$ and $\csc^2 \phi = 9$.
**Step 2:** The value of $m$ is calculated using the expression:
$$m = (1 + \sec^2\theta) + (1 + \csc^2\phi)$$
Substituting the squared values of $\sec \theta$ and $\csc \phi$:
$$m = (1 + 4) + (1 + 9)$$
$$m = 5 + 10$$
$$m = 15$$
**Step 3:** We need to find the value of $m^2 + m + 10$.
Substituting $m=15$ into the expression:
$$m^2 + m + 10 = (15)^2 + 15 + 10$$
$$m^2 + m + 10 = 225 + 15 + 10$$
$$m^2 + m + 10 = 250$$
Correct Answer: 250