O"Proving a Trig Identity: Sec^6x-Tan^6x = 1+3Sec^2xTan^2x | Tips & Tricks

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The discussion focuses on proving the trigonometric identity sec^6 x - tan^6 x = 1 + 3sec^2 x tan^2 x. Initial attempts involved factoring the left-hand side and applying Pythagorean identities, but progress was slow. A helpful hint suggested using the expansion of (tan²x + 1)³ - tan^6 x, which led to a breakthrough. After following the hint, the participant successfully demonstrated that the left-hand side simplifies to the right-hand side. The exchange highlights the importance of collaborative problem-solving in understanding trigonometric identities.
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Homework Statement



Show that the LHS can be changed into the RHS.
sec^6 x-tan^6x=1+3sec^2x tan^2x


Homework Equations



Trig identities.

The Attempt at a Solution


I tried factoring the LHS:
(sec^2-tan^2)(sec^4+sec^2tan^2+tan^4)
sec^2-tan^2=1 so that leaves me with the other thing in the parentheses. I have tried using the Pythagorean identities on sec^2 and tan^2, I have broken up the sec^4 into sec^2*sec^2...

I just am not getting anywhere.
Pointers would be nice.
CC
 
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i get the RHS le.
try (tan²x+1)³ - tan^6 x
then you expand.
Cheers:smile:

i will be offline , i shall leave the ans in the spoiler (:

next step:
3tan²x+3tan^4 x +1
next:
1+3tan²x(tan²x+1)
Finally:
1+3sec²tan²x
=RHS(shown)
 
Last edited:
I got it easily after the first hint. Thanks! I had been staring at it too long to see another way.
 

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