Prove the identity Factor the left side of the identity Appl

Prove the identity. Factor the left side of the identity. Apply a Pythagorean identity to rewrite the above expression entirely in terms of tan theta and simplify each factor. Do not multiply.

Solution

given 2 tan²(x) sec²(x) - sec⁴(x) = tan⁴(x) -1

factoring left hand side

  sec⁴(x) can be written as sec²(x) * sec²(x)

so left hand side can be written as 2 tan²(x) sec²(x) - sec²(x) * sec²(x)

  sec²(x) [ 2 tan²(x) - sec²(x) ]

but the identity sec²(x) - tan²(x)=1

so plug sec²(x) =  tan²(x)+1

now left hand side becomes =   sec²(x) [ 2 tan²(x) - tan²(x) -1 ]

=   sec²(x) [ tan²(x) -1 ]

factoring = [ 1 + tan²(x) ]  [ tan²(x) -1 ]

now to prove it is equal to the right hand side

just multiply these two  [ 1 + tan²(x) ]  [ tan²(x) -1 ] =   tan²(x) -1 +tan⁴(x) -tan²(x)

= -1 +tan⁴(x)

= tan⁴(x) -1

this is exactly equals to the right hand side

so hence proved

pythagorem identity is

[ 1 + tan²(x) ]  [ tan²(x) -1 ] = tan⁴(x) -1

it is a pythagorem identity