My book says that if I learn the derivatives of sin, tangent, and secant function then I can find the rest of the trig derivatives. To do this I must replacing each function by its coresponding and put a negative on the right hand side of the new derivative equation.So I just want to make sure I have these three basic proofs down.

I can do the proofs for SinX using the f'(x) limit formula and the Addition formula for sin(a+b)...

I find the tanX by rewriting it as

and using the quotient rule...

But when I try to find the derivative of the sec I ran into a little issue as far as factoring and rewriting

Quotient rule

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My first instinct was to take out the sinX

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But this is wrong and I am suppose to take out

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This has me frustrated. Am I supposed to avoid a squared trig value unless its for the derivatives of tan and cot.