Question

prove that sec^6-tan^6=1+3sec^2xtan^2x

Answer 1

Solution :

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We know the algebraic identity:

a³ - b³ - 3ab( a - b ) = ( a - b )³

Or

a³ - b³ = ( a - b )³ + 3ab( a - b )

and

we know the Trigonometric identity :

Sec²x - tan² x = 1

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Here

LHS = sec^6 x - tan^6 x

= ( sec²x )³ - ( tan²x )³

= ( sec²x-tan²x)³+3sec²xtan²x(sec²x-tan²x)

= 1 + 3sec²xtan²x

=RHS

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