What is cot² (sec-12) + tan 2 (cosec-13) equal to?

(a) 11/12

(b) 11/24

(c) 7/24

(d) 1/24

Let's evaluate the given expression step by step:

The expression is:
\[
\cot^2(\sec^{-1}(2)) + \tan^2(\csc^{-1}(3))
\]

### Step 1: Evaluate \(\sec^{-1}(2)\)
The secant inverse of 2 means that the secant of an angle, say \(\theta\), is 2. Recall that:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
Thus, we have:
\[
\sec(\theta) = 2 \implies \cos(\theta) = \frac{1}{2}
\]
This implies that \(\theta = \sec^{-1}(2)\) corresponds to an angle whose cosine is \(\frac{1}{2}\), which is \(\theta = 60^\circ\) or \(\theta = \frac{\pi}{3}\) radians.

Now, we need to find \(\cot^2(\theta)\):
\[
\cot^2(\theta) = \left(\frac{1}{\tan(\theta)}\right)^2
\]
For \(\theta = \frac{\pi}{3}\), we know \(\tan(\frac{\pi}{3}) = \sqrt{3}\), so:
\[
\cot^2(\frac{\pi}{3}) = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}
\]

### Step 2: Evaluate \(\csc^{-1}(3)\)
The cosecant inverse of 3 means that the cosecant of an angle, say \(\alpha\), is 3. Recall that:
\[
\csc(\alpha) = \frac{1}{\sin(\alpha)}
\]
Thus, we have:
\[
\csc(\alpha) = 3 \implies \sin(\alpha) = \frac{1}{3}
\]
This implies that \(\alpha = \csc^{-1}(3)\).

Now, we need to find \(\tan^2(\alpha)\):
\[
\tan^2(\alpha) = \frac{\sin^2(\alpha)}{\cos^2(\alpha)}
\]
Using \(\sin(\alpha) = \frac{1}{3}\), we can find \(\cos^2(\alpha)\) using the Pythagorean identity:
\[
\sin^2(\alpha) + \cos^2(\alpha) = 1 \implies \cos^2(\alpha) = 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9}
\]
Now, we can find \(\tan^2(\alpha)\):
\[
\tan^2(\alpha) = \frac{\left(\frac{1}{3}\right)^2}{\frac{8}{9}} = \frac{\frac{1}{9}}{\frac{8}{9}} = \frac{1}{8}
\]

### Step 3: Combine the results
Now that we have both values, we can add them together:
\[
\cot^2(\sec^{-1}(2)) + \tan^2(\csc^{-1}(3)) = \frac{1}{3} + \frac{1}{8}
\]
To add these, we need a common denominator:
\[
\frac{1}{3} + \frac{1}{8} = \frac{8}{24} + \frac{3}{24} = \frac{11}{24}
\]

Thus, the value of the expression is:
\[
\boxed{\frac{11}{24}}
\]

### Answer:
The correct option is **(b) 11/24**.

---
Do you have any questions or want more details on this solution? Here are 5 related questions for practice:

1. What is \(\cot^2(\sec^{-1}(3)) + \tan^2(\csc^{-1}(4))\)?
2. Find \(\sec(\cos^{-1}(1/2)) + \csc(\sin^{-1}(1/3))\).
3. How do you simplify \(\cot(\tan^{-1}(x))\)?
4. What is \(\cot^2(60^\circ) + \tan^2(30^\circ)\)?
5. How do you solve for \(\theta\) if \(\sec(\theta) = 2\) and \(\csc(\theta) = 3\)?

**Tip:** Memorizing the values of trigonometric functions at common angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\) can help you solve problems quickly.

What is cot² (sec-12) + tan² (cosec-13) equal to?
(a) 11/12
(b) 11/24
(c) 7/24
(d) 1/24

Discover how to solve cot²(sec⁻¹(2)) + tan²(cosec⁻¹(3)) in this detailed trigonometry tutorial. Learn key concepts like inverse trigonometric functions, Pythagorean identities, and cotangent and tangent squared values. Perfect for Grades 10-12 students.

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