Question

Can someone pls do 25 and 26 with working

Answer 1

25.

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`(\tan x + \sin x)/(\tan x - \sin x) = (\sec x + 1)/(\sec x - 1)`

`(\tan x + \sin x)/(\tan x - \sin x)`

`= ((\sin x)/(\cos x) + \sin x)/((\sin x)/(\cos x) - \sin x)`

`= (\sin x ( (1)/(\cos x) + 1))/(\sin x( ( 1)/(\cos x) - 1))`

`= (\sec x + 1)/(\sec x - 1) \quad\checkmark`

26.

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`(\cot \theta + \csc \theta - 1)/(\cot \theta - \csc \theta +1) = (1+\cos \theta)/(\sin \theta)`

That's the cotangent half angle formula on the right, so I guess on the left too.

I fooled around with this one for a while before I took the hint which was to let
`\theta=2x`.

`(\cot 2x + \csc 2x - 1)/(\cot 2x- \csc 2x +1)`

`=( (\cos 2x / \sin 2x) + (1/\sin 2x ) - 1)/(\cos 2x/\sin 2x - 1/\sin 2x + 1)`

`=(\cos 2x + 1 - \sin 2x)/(\cos2x - 1 + \sin 2x)`

`=(2\cos^2 x - 1 + 1 -2 \cos x \sin x )/(1 - 2\sin^2x-1 + 2\sin x \cos x)`

`=(2 \cos^2 x -2 \cos x \sin x )/(-2\sin^2x+2 \sin x \cos x)`

`=(2\cos x)/(2 \sin x) \cdot (\cos x- \sin x)/(-\sin x +\cos x)`

`=(2\cos x)/(2\sin x)`

`=(2\cos^2 x)/(2\sin x\cos x)`

`=(1 + 2\cos^2 x - 1)/(2\sin x\cos x)`

`= (1 + \cos 2x)/(\sin 2x)`

`=(1+\cos \theta)/(\sin \theta) \quad\checkmark`

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For another answer, let's use the hint on this one, which was to write

`1=\csc^2 \theta - \cot^2 \theta`

That's a good hint; first let's verify if it's true.

`\sin^2 \theta + \cos^2 \theta = 1`

`\sin^2 \theta = 1 - \cos^2 \theta`

`1 = (1)/(\sin ^2 \theta) - (\cos ^2 \theta)/(\sin ^2 \theta)`

`1 = \csc^2 \theta - \cot^2 \theta \quad\checkmark`

Now,

`(\cot \theta + \csc \theta - 1)/(\cot \theta - \csc \theta +1)`

`= (\cot \theta + \csc \theta - (\csc^2 \theta - \cot^2 \theta))/(\cot \theta - \csc \theta +1)`

`= (\cot \theta + \csc \theta +(\cot^2 \theta - \csc^2 \theta))/(\cot \theta - \csc \theta +1)`

`= (\cot \theta + \csc \theta +(\cot \theta - \csc \theta)(\cot \theta + \csc \theta))/(\cot \theta - \csc \theta +1)`

`= ((\cot \theta + \csc \theta)(1+ \cot \theta - \csc \theta))/(\cot \theta - \csc \theta +1)`

`=\cot \theta + \csc \theta`

`=( \cos \theta)/(\sin \theta)+ (1)/(\sin \theta)`

`=(1+ \cos \theta)/(\sin \theta) \quad\checkmark`