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1 Pure Mathematics

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Question 298
a.

Show that the trigonometric identity

sin⁡2x1+tan⁡2x≡2sin⁡xcos⁡3x \frac{\sin 2x}{1 + \tan^2 x} \equiv 2 \sin x \cos^3 x 1+tan2xsin2x​≡2sinxcos3x

is valid for all x x\,x where the expression is defined.

[3]
b.

In a wave mechanics simulation, the rate of change of energy density E E\,E with respect to phase ϕ\phiϕ (in radians) is modeled by the equation:

dEdϕ=30sin⁡6ϕ1+tan⁡23ϕ \frac{dE}{d\phi} = \frac{30 \sin 6\phi}{1 + \tan^2 3\phi} dϕdE​=1+tan23ϕ30sin6ϕ​

Hence, determine the general expression for E(ϕ)E(\phi)E(ϕ).

[6]
Markscheme

1 Pure Mathematics Questions

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