Screenie 2 1 2 X 4

• X 2 3 X 4 1
• 1 2 8 4 X
• Screenie 2 1 2 X 4.5
• Screenie 2 1 2 X 4

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Proofs of Derivative of Trig Functions

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Given: lim(d->0) sin(d)/d = 1.
Solve:

sin(x) = lim_(d->0) ( sin(x+d) - sin(x) ) / d

= lim ( sin(x)cos(d) + cos(x)sin(d) - sin(x) ) / d

= lim ( sin(x)cos(d) - sin(x) )/d + lim cos(x)sin(d)/d

= sin(x) lim ( cos(d) - 1 )/d + cos(x) lim sin(d)/d

= sin(x) lim ( (cos(d)-1)(cos(d)+1) ) / ( d(cos(d)+1) ) + cos(x) lim sin(d)/d

= sin(x) lim ( cos²(d)-1 ) / ( d(cos(d)+1 ) + cos(x) lim sin(d)/d

X 2 3 X 4 1

= sin(x) lim -sin²(d) / ( d(cos(d) + 1) + cos(x) lim sin(d)/d

= sin(x) lim (-sin(d)) * lim sin(d)/d * lim 1/(cos(d)+1) + cos(x) lim sin(d)/d

= sin(x) * 0 * 1 * 1/2 + cos(x) * 1 = cos(x)Q.E.D.

Proof of cos(x) : from the derivative of sine

This can be derived just like sin(x) was derived or more easily from the result of sin(x)

Colibri 1 9 100. Given: sin(x) = cos(x); Chain Rule.
Solve:

cos(x) = sin(x + PI/2)

cos(x) = sin(x + PI/2)

= sin(u) * (x + PI/2) (Set u = x + PI/2)

= cos(u) * 1 = cos(x + PI/2) = -sin(x)Q.E.D.

Proof of tan(x) : from the derivatives of sine and cosine

Given: sin(x) = cos(x); cos(x) = -sin(x); Quotient Rule.
Solve: Optimage advanced image optimization tool 3 3 1.

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

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

= ( cos(x) sin(x) - sin(x) cos(x) ) / cos²(x)

= ( cos(x)cos(x) + sin(x)sin(x) ) / cos²(x)

= 1 + tan²(x) = sec²(x)Q.E.D.

Proof of csc(x), sec(x), cot(x) : from derivatives of their reciprocal functions

Given: sin(x) = cos(x); cos(x) = -sin(x); tan(x) = sec²(x); Quotient Rule.
Solve:

1 2 8 4 X

csc(x) = 1/sin(x) = ( sin(x) (1) - 1 sin(x) ) / sin²(x) = -cos(x) / sin²(x) = -csc(x)cot(x)

Screenie 2 1 2 X 4.5

sec(x) = 1/cos(x) = ( cos(x) (1) - 1 cos(x) ) / cos²(x) = sin(x) / cos²(x) = sec(x)tan(x)

Screenie 2 1 2 X 4

cot(x) = 1/tan(x) = ( tan(x) (1) - 1 tan(x) ) / tan²(x) = -sec²(x) / tan²(x) = -csc²(x)Q.E.D.