In this discussion, we are going to verify trigonometric identities using the double-angle identities. We will also be using other trig identities that we have discussed in previous readings.
Here is an outline of this discussion:
Brief Review of the Fundamental Trig Identities
Verify that (1-cos2x)/sin2x = tanx
Verify that sin2x = 2tanx/(1+ tan^2x)
Let’s begin!
Brief Review of the Fundamental Trig Identities
Verify that (1-cos2x)/sin2x = tanx
We begin with the left-hand side of the equation and slowly manipulate it until it looks like the right-hand side.
LHS
= (1-cos2x)/sin2x
= (1-(1-2sin^2x)) / 2sinxcosx
= (1-1+2sin^2x) / 2sinxcosx
= 2sin^2x / 2sinxcosx
= sinx / cosx
= tanx
= RHS
Verify that sin2x = 2tanx/(1+ tan^2x)
This time, let us begin with the right-hand side and slowly manipulate it until it looks like the left-hand side.
RHS
= 2tanx/(1+ tan^2x)
= (2sinx / cosx )/ (sec^2x)
= (2sinx / cosx )/ (1 / cos^2x)
= 2sinxcos^2x / cosx
= 2sinxcosx
= sin2x
= LHS
We are done!
Thank you for reading! I hope that I can assist you on your math journey!
Sincerely,
➕MathAngel369➕
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