Help w/ the difference quotient

EL ALEM
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Homework Statement


Show that if f(x)=sinx then (f(x+h)-f(x))/h=((sin(h/2))/(h/2))(cos(x+h/2)


Homework Equations


Trig identities, possibly the half angle formulas?


The Attempt at a Solution


(f(x+h)-f(x))/h
= (f(x+ h/2 + h/2)-f(x))/(h/2 + h/2)
= (sin(x+ h/2 + h/2)-sin(x))/(h/2 + h/2)
im stuck after this, don't know what to do..
 
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Ok so I used the sum to product identity and ended up w/ this

(2sin(h/2)cos((2x+h)/2))/h

not sure how to simplify it further

NM got it.
 
Last edited:
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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