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Home » DDC Codes Exposed: The Secret Shortcuts Every Tech Guru Needs! - Databee Business Systems

DDC Codes Exposed: The Secret Shortcuts Every Tech Guru Needs! - Databee Business Systems

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 5Question: Find the minimum value of $(\cos x + \sec x)^2 + (\sin x + \csc x)^2$ for $0 < x < \frac{\pi}{2}$.

📌 We begin by expanding the expression:

📌 Recall that $\cos x \sec x = 1$ and $\sin x \csc x = 1$, so the expression becomes:

📌 \cos^2 x + \sin^2 x + 2 + \sec^2 x + \csc^2 x + 2

📌 Since $\cos^2 x + \sin^2 x = 1$, this simplifies to:

📌 + 2 + \sec^2 x + \csc^2 x + 2 = 5 + \sec^2 x + \csc^2 x

📌 Now, write $\sec^2 x = 1 + \tan^2 x$ and $\csc^2 x = 1 + \cot^2 x$:

📌 + (1 + \tan^2 x) + (1 + \cot^2 x) = 7 + \tan^2 x + \cot^2 x

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Question:** What is the remainder when \( 1001 + 1003 + 1005 + 1007 +...
Solution:** Compute the sum:
+ 1003 + 1005 + 1007 + 1009 = 5 \cdot 1005 = 5025
Now compute \( 5025 \mod 11 \). Use alternating sum for modulo 11:
\( 5 - 0 + 2 - 5 = 2 \), or directly:
\( 5025 \div 11 \): \( 11 \times 456 = 5016 \), so \( 5025 - 5016 = 9...
Alternatively, each term mod 11:
\( 1001 \equiv 0 \pmod{11} \) (since \( 11 \times 91 = 1001 \))
\( 1003 \equiv 2 \)
\( 1005 \equiv 4 \)

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