SOLUTION: How to establish the identities: (sin(a-b))/(sin(a)cos(b))=1-cot(a)tan(b) and 2sin(2theta)(1-2sin^2theta)=sin(4theta)

Question 608443: How to establish the identities:
(sin(a-b))/(sin(a)cos(b))=1-cot(a)tan(b)
and
2sin(2theta)(1-2sin^2theta)=sin(4theta)
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
How to establish the identities:
(sin(a-b))/(sin(a)cos(b))=1-cot(a)tan(b)
Use addition formula for sin: sin(a-b)=sina*cosb-cosa*sinb
Start with left side:
(sin(a-b))/(sina*cosb)
=(sina*cosb-cosa*sinb)/(sina*cosb)
=1-(cosa*sinb)/(sina*cosb)
=1-cota*tanb
verified: left side=right side
..
and
2sin(2theta)(1-2sin^2theta)=sin(4theta)
use x for theta
Start with right side:
sin4x=2sin2xcos2x
=2sin2x(cos^2x-sin^2x)
=2sin2x(1-sin2^x-sin^2x)
=2sin2x(1-2sin^2x)
verified: right side=left side
RELATED QUESTIONS
Show the following identities. A). {{{cosx+sinx/cosx-sinx - cosx-sinx/cosx+sinx =... (answered by MathLover1)

which of the following is equivalent to {{{(1-Cos^2theta)/(Cos^2theta)}}}?... (answered by Edwin McCravy)

How can I prove this identity? (1-cos^2theta)(1+cos^2theta)=2sin^2theta-sin^4theta? I... (answered by Alan3354,scott8148)

Suppose that sin(theta) = 6/10 and theta is in the second quadrant. Use trigonometric... (answered by Alan3354)

Prove the following identity: cos^4theta + 2cos^2theta sin^2theta + sin^4theta =... (answered by sarah_adam)

Show that (a) sin 2theta * cos theta + cos 2theta * sin theta = sin 4theta * cos theta (answered by mccravyedwin)

cos^2theta sectheta= A. tan theta B. cot theta C. sin theta D. cos theta E. csc theta... (answered by texttutoring)

Find all solutions to the equation on the interval [0,2pie) cos(2theta)-tan(theta)=1... (answered by jim_thompson5910)

cos^2theta-sin^2theta/cos^2theta+sin theta cos theta=1-tan... (answered by greenestamps)