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Home » 3x + 4x + 5x = 60 \\ - Coaching Toolbox

3x + 4x + 5x = 60 \\ - Coaching Toolbox

📅 May 4, 2026 👤 scraface

May 04, 2026

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Let \(u = \sin^2 x\), then \(\cos^2 x = 1 - u\), so:
rac{u}{1 - u} + rac{1 - u}{u} + 2 = rac{u^2 + (1 - u)^2}{u(1 - u)} +...
But better: use identity:
an x + \cot x = rac{\sin x}{\cos x} + rac{\cos x}{\sin x} = rac{\si...
Now the full expression:
+ \sin 2x + rac{4}{\sin^2 2x}
Let \(t = \sin 2x\), \(0 < t \leq 1\) (since \(x \in (0, \pi/2)\Righta...
Define \(f(t) = 1 + t + rac{4}{t^2}\), \(0 < t \leq 1\)
Derivative: \(f'(t) = 1 - rac{8}{t^3}\)
Set to zero: \(1 = rac{8}{t^3} \Rightarrow t^3 = 8 \Rightarrow t = 2\...

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