Final answer:
To simplify the expression tan(π/2 - x)csc(x)/csc²x, we can use trigonometric identities to rewrite the expression in terms of sine and cosine. The simplified expression is cos(x) - sin(x).
Step-by-step explanation:
To simplify the expression tan(π/2 - x)csc(x)/csc²x, we can use trigonometric identities to rewrite the expression in terms of sine and cosine.
First, we know that csc(x) is equal to 1/sin(x). So the expression becomes:
tan(π/2 - x) * (1/sin(x))/(1/sin²(x))
Next, we can simplify the tan(π/2 - x) using the identity tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b)). In this case, a = π/2 and b = -x. So:
tan(π/2 - x) = (tan(π/2) + tan(-x))/(1 - tan(π/2)tan(-x)) = (1 + tan(-x))/(1 - tan(-x))
Now, substituting this back into the expression, we get:
(1 + tan(-x))/(1 - tan(-x)) * (1/sin(x))/(1/sin²(x))
Using the identity tan(-x) = -tan(x), we can further simplify:
(1 - tan(x))/(1 + tan(x)) * sin²(x)
Finally, we can use the identity sin²(x) = 1 - cos²(x) to rewrite the expression as:
(1 - tan(x))/(1 + tan(x)) * (1 - cos²(x))
Expanding the numerator and denominator, we get:
[1 - tan(x)][1 - cos²(x)]/(1 + tan(x))
Simplifying further:
[1 - tan(x)][sin²(x)]/[(1 + tan(x))(cos²(x))]
Finally, using the identity sin²(x) = 1 - cos²(x), the expression can be simplified to:
[1 - tan(x)]/(1 + tan(x))
This can be rewritten as [1 - sin(x)/cos(x)]/(1 + sin(x)/cos(x)). Multiplying the numerator and denominator by cos(x), we get (cos(x) - sin(x))/cos(x)cos(x) = (cos(x) - sin(x))/cos²(x) = cos(x)/cos²(x) - sin(x)/cos²(x) = cos(x) - sin(x).