Prove that sinxtanx=1/cosx - cosx \sin(x) \tan(x) = (1)/( \cos(x) ) - \cos(x) ​

Question

Prove that sinxtanx=1/cosx - cosx


`\sin(x) \tan(x) = (1)/( \cos(x) ) - \cos(x)`
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Answer 1

See below

Explanation:

We want to prove that

`\sin(x)\tan(x) = (1)/(\cos(x)) - \cos(x), \forall x \in\mathbb{R}`

Taking the RHS, note

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

Remember that

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

Therefore,

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

Once

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

Then,

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

Hence, it is proved