Can we write $\cos \theta - 2 \cos 3 \theta + \cos 5\theta$ using $\sin$?

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1. Prove the identity \[\cos \theta - 2 \cos 3 \theta + \cos 5\theta = 2\sin \theta (\sin 2 \theta - \sin 4 \theta).\]

Can we combine two \(\cos\) terms using an expression for the difference of two cosines?

We could use this twice here…

2. Solve the equations

   1. \(\cos 2x = \sin x\),

   2. \(3 \sec^2 x = \tan x + 5\)

   giving in each case all solutions between \(0^\circ\) and \(360^\circ\).

Can we write the first equation just in terms of \(\sin x\)?

How can we relate \(\sec^2 x\) to \(\tan x\)?

This would reduce the equations to quadratic ones…