Calculate without a calculator \( \frac{5}{\tan ^{2} A} \)\( \frac{\frac{7}{\cos ^{2} B}}{} \)

We start with the expression \[ \frac{5}{\tan^2 A} \cdot \frac{7}{\cos^2 B}. \] **Step 1. Multiply the fractions.** Multiplying the numerators and denominators gives \[ \frac{5 \cdot 7}{\tan^2 A \cdot \cos^2 B} = \frac{35}{\tan^2 A \cos^2 B}. \] **Step 2. Express \(\tan^2 A\) in terms of sine and cosine.** Recall that \[ \tan A = \frac{\sin A}{\cos A} \quad \Rightarrow \quad \tan^2 A = \frac{\sin^2 A}{\cos^2 A}. \] Then, substitute into the expression: \[ \frac{35}{\frac{\sin^2 A}{\cos^2 A}\cos^2 B}. \] **Step 3. Simplify by multiplying by the reciprocal.** Multiply numerator and denominator by \(\cos^2 A\): \[ \frac{35 \cos^2 A}{\sin^2 A\, \cos^2 B}. \] **Step 4. Rewrite in alternative forms.** We can recognize that \[ \frac{\cos^2 A}{\sin^2 A} = \cot^2 A \quad \text{and} \quad \frac{1}{\cos^2 B} = \sec^2 B. \] Thus, the expression becomes \[ 35 \cot^2 A\, \sec^2 B. \] **Final Answer:** \[ 35 \cot^2 A\, \sec^2 B. \]