simplify the expression.
tan(π/2-x)tanx
tan (pi/2 - x) = sin (pi/2 - x) / cos (pi/2 - x)
But sin (pi/2 - x) = cos x
and
cos (pi/2 - x) = sin x
<=>
tan (pi/2 - x) = cos x / sin x = cotan x
<=>
tan (pi/2 - x) * tan x =
cotan x * tan x =
(cos x / sin x) * (sin x / cos x) =
1
works for me.
the co-functions are the functions of the complementary angles. So, by definition, tan(π/2-x) = cot(x). Your proof works as well, though.
To simplify the expression tan(π/2 - x)tan(x), we can use the trigonometric identity for the tangent of the difference of two angles:
tan(A - B) = (tanA - tanB) / (1 + tanA * tanB)
Let's apply this identity to the given expression:
tan(π/2 - x)tanx = (tan(π/2) - tan(x)) / (1 + tan(π/2) * tan(x))
But the tangent of π/2 is undefined, so we can simplify further:
= (undefined - tan(x)) / (1 + undefined * tan(x))
Since we have an undefined value, we cannot simplify it any further. The expression tan(π/2 - x)tan(x) remains as it is.