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Home » 10 Hidden Secrets of the Waco Series You’ve Never Heard Before! - jntua results

10 Hidden Secrets of the Waco Series You’ve Never Heard Before! - jntua results

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Number of doubling periods: $ 10 / 2 = 5 $.

📌 Final cases = $ 10 \times 2^5 = 10 \times 32 = 320 $.

📌 #### 320Question: Find the minimum value of $(\sin x + \csc x)^2 + (\cos x + \sec x)^2$.

📌 Solution: Expand the expression: $(\sin x + \csc x)^2 = \sin^2 x + 2 + \csc^2 x$ and $(\cos x + \sec x)^2 = \cos^2 x + 2 + \sec^2 x$. Combine terms: $\sin^2 x + \cos^2 x + 4 + \csc^2 x + \sec^2 x$. Since $\sin^2 x + \cos^2 x = 1$, this simplifies to $5 + \csc^2 x + \sec^2 x$. Rewrite $\csc^2 x = 1 + \cot^2 x$ and $\sec^2 x = 1 + \tan^2 x$, so total becomes $7 + \tan^2 x + \cot^2 x$. Let $t = \tan^2 x$, then expression is $7 + t + \frac{1}{t}$. The minimum of $t + \frac{1}{t}$ for $t > 0$ is $2$ by AM-GM inequality. Thus, the minimum value is $7 + 2 = \boxed{9}$.

📌 Question: Find the matrix $\mathbf{M}$ such that $\mathbf{M} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$.

📌 Solution: Let $\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Multiply $\mathbf{M}$ with $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ to get:

📌 $a + 3b = 5$

📌 $2a + 4b = 6$

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