Solving $\tan(7x)=\cot(8x)$ – Step‑by‑Step 2.5D Motion Graphic

Overview

A concise visual walk‑through of solving the trigonometric equation $\tan(7x)=\cot(8x)$. The animation demonstrates the key identity conversion, algebraic manipulation, and final isolation of $x$, highlighting each transformation with high‑contrast 2.5D minimalist graphics.

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Phases

#	Phase Name	Duration	Description
1	The Challenge	~5 s	Dark midnight‑blue canvas fades in, the equation $\tan(7x)=\cot(8x)$ writes itself in crisp white, a neon‑orange question mark pulses above, and a "How do we solve for x?" card bounces in the lower third.
2	Identity Key	~7 s	Camera zooms toward the $\cot(8x)$ term. The term flashes with a cyan outline, a holographic card slides up displaying the co‑function identity $\cot\theta=\tan(90^{\circ}\! -\theta)$. The term morphs fluidly into $\tan(90^{\circ}\! -8x)$.
3	Equation Alignment	~8 s	Camera pans slightly left to recenter. The full equation updates to $\tan(7x)=\tan(90^{\circ}\! -8x)$. Both "tan" prefixes glow orange then fade, leaving only the angles. Angles drop down and align as $7x = 90^{\circ} - 8x$.
4	Algebraic Shift	~8 s	The "‑ 8x" term detaches, slides across the equals sign, flips sign to "+" and changes to electric cyan. It settles next to $7x$ forming $7x + 8x = 90^{\circ}$. The two terms fuse in a brief energy burst, becoming $15x = 90^{\circ}$.
5	Final Isolation	~7 s	The coefficient "15" separates, slides under "90°" as a fraction bar appears, yielding $x = \frac{90^{\circ}}{15}$. A digital ring spins around the fraction for 1.5 s, then collapses to the final answer $x = 6^{\circ}$. The screen dims, the answer scales up 20 % and glows neon orange, followed by a minimalist geometric confetti burst.
6	Outro	~2 s	Fade out leaving the final answer briefly centered before the scene ends.

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Layout

┌─────────────────────────────────────────────┐
│                 TITLE AREA                  │
├──────────────────────┬──────────────────────┤
│                      │                      │
│      MAIN VISUAL     │   SUPPORT PANEL      │
│   (equation & steps)│ (identity card,      │
│                      │  occasional notes)  │
├──────────────────────┴──────────────────────┤
│                BOTTOM CAPTION               │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Title	Optional short title (e.g., "Solve $\tan(7x)=\cot(8x)$")	Fades in at start of Phase 1, fades out with Outro
Main Visual	Central equation and all step‑by‑step transformations (2.5D layers, neon highlights)	Primary focus; occupies most of the screen width
Support Panel	Holographic formula card in Phase 2, occasional brief label arrows	Slides in/out as needed; background transparent
Bottom Caption	Small white caption for phase labels (e.g., "Identity conversion")	Appears briefly at start of each phase, then fades

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Notes

• All colors follow the palette: background midnight blue (#0D1B2A), cyan accents (#00B4D8), neon orange focus (#FF4D00), white text (#FFFFFF).
• Camera motions are limited to orthographic zoom/pan; no perspective distortion.
• Visual effects (glow, pulse, fade, sign flip) are kept short to keep total runtime under 35 seconds.
• No textual narration is required; all information is conveyed visually.
• The specification assumes a single Manim Scene class; all phases are orchestrated sequentially within that scene.
• Minimalist geometric confetti uses simple polygons (triangles, squares) in cyan and orange, appearing only in the final 0.5 s.