In this Discussion, we are going to Verify more challenging Trig Identities Using the Pythagorean Identities.
Here is an outline of this article:
- Review of the Fundamental Trig Identities
- Verifying ( tanx + cotx )^2 = sec^2x +csc^2x
Okay, let’s begin!
Review of the Fundamental Trig Identities
Here is a quick review of the fundamental trig identities.
Verifying ( tanx + cotx )^2 = sec^2x +csc^2x
Now we are ready to verify are trig identity.
Here, we have
( tanx + cotx )^2 = sec^2x +csc^2x
Begin with the left-hand side (LHS) and slowly manipulate until it looks like the right-hand side (RHS).
LHS = ( tanx + cotx )^2
=
( tanx + cotx )( tanx + cotx )
Apply the FOIL Method
tan^2x + 2tanxcotx + cot^2x
=
Use the quotient identity
sin^2x/cos^2x + 2(sinx/cosx)*(cosx/sinx) + cos^2x/sin^2x
Cancel common factors
sin^2x/cos^2x + 2 + cos^2x/sin^2x
Rewrite
sin^2x/cos^2x + 1 + 1 + cos^2x/sin^2x
sin^2x/cos^2x + cos^2x / cos^2x + sin^2x/ sin^2x + cos^2x/sin^2x
Use a Pythagorean Identity
1/ cos^2x + 1/ sin^2x
sec^2x +csc^2x
=
RHS
We have shown that the left hand side is equal to the right hand side.
Thank you for reading!
I hope that I can assist you on your math journey!
Sincerely,
MathAngel369
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