Final answer:
To verify the identity tan(t) + cot(t) = sec(t)csc(t), express tan(t) and cot(t) in terms of sin(t) and cos(t). Substitute these expressions into the left side of the equation and simplify using trigonometric identities. The identity is verified when the left side is equal to the right side.
Step-by-step explanation:
To verify the identity tan(t) + cot(t) = sec(t)csc(t), we need to manipulate the left side of the equation to match the right side. Let's start by expressing tan(t) and cot(t) in terms of sin(t) and cos(t):
tan(t) = sin(t)/cos(t) and cot(t) = cos(t)/sin(t).
Now substitute the above expressions into the left side of the equation:
sin(t)/cos(t) + cos(t)/sin(t).
To simplify this expression, find a common denominator and combine the fractions:
(sin(t)sin(t) + cos(t)cos(t))/(cos(t)sin(t)).
Using the Pythagorean identity sin²(t) + cos²(t) = 1, we can simplify the numerator to 1 and the denominator to sin(t)cos(t):
1/(cos(t)sin(t)) = sec(t)csc(t).
Therefore, the identity tan(t) + cot(t) = sec(t)csc(t) is verified.