证明:(tan^2)x-(sin^2)x=(tan^2)x(sin^2)x
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证明:(tan^2)x-(sin^2)x=(tan^2)x(sin^2)x
Notice that
tan²(x) = sin²(x) / cos²(x) .
Using this on the right-hand side of the equation and gathering into a common fraction,we get
tan²(x) - sin²(x) = (sin²(x) / cos²(x)) - sin²(x)
= (sin²(x) - sin²(x) cos²(x)) / cos²(x)
= (sin²(x) (1 - cos²(x))) / cos²(x) .
Recalling the Pythagorean identity
sin²(x) + cos²(x) = 1 ,
we get
(sin²(x) (1 - cos²(x))) / cos²(x) = (sin²(x) sin²(x)) / cos²(x) .
= sin²(x) (sin²(x) / cos²(x))
= sin²(x) tan²(x) ,
just as we wanted.
tan²(x) = sin²(x) / cos²(x) .
Using this on the right-hand side of the equation and gathering into a common fraction,we get
tan²(x) - sin²(x) = (sin²(x) / cos²(x)) - sin²(x)
= (sin²(x) - sin²(x) cos²(x)) / cos²(x)
= (sin²(x) (1 - cos²(x))) / cos²(x) .
Recalling the Pythagorean identity
sin²(x) + cos²(x) = 1 ,
we get
(sin²(x) (1 - cos²(x))) / cos²(x) = (sin²(x) sin²(x)) / cos²(x) .
= sin²(x) (sin²(x) / cos²(x))
= sin²(x) tan²(x) ,
just as we wanted.
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