I need help finding the derivative of f(x)=secx^2 using the double angle formula. I know that the answer is 2xsec(x^2)tan(x^2) but I don't understand how they got the answer.
Thank you for the opportunity to help you with your question!
Notice how a "co-(something)"
trig ratio is always the reciprocal of some "non-co" ratio.
You can use this fact to help you keep straight that cosecant goes with
sine and secant goes with cosine.
+ cos2(t) = 1 tan2(t)
+ 1 = sec2(t) 1
+ cot2(t) = csc2(t)
The above, because they involve squaring
and the number 1, are the "Pythagorean" identities. You can
see this clearly if you consider the unit circle, where sin(t)
= y, cos(t)
= x, and the hypotenuse is 1.
= –sin(t) cos(–t)
= cos(t) tan(–t)
Notice in particular that sine and tangent
are odd function, while cosine is an even function
angle sum and -Difference Identities
+ β) = sin(α)cos(β) + cos(α)sin(β)
β) = sin(α)cos(β) – cos(α)sin(β)
β) = cos(α)cos(β) – sin(α)sin(β)
β) = cos(α)cos(β) + sin(α)sin(β)
double angle Identities
= cos2(x) – sin2(x)
= 1 – 2sin2(x) = 2cos2(x)
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