need help simplifying some trig identities such as:

(csc t) (sin t)
cos t = 0.75 and tan t = 0.88, find sec t and cot t;

(cot2t) (sin2t) + sin2 t;

csc2t - cot2t/sin2t

small discrepancy in your given information

if cos t = .75 = 75/100 = 3/4
then we have a triangle with x=3, y = ? and hypotenuse = 4
3^2 + y^2 = 4^2
y^2 = 7
y = √7
then tan t = √7/3 = .881917.. , not .88
so sec t = 4/3 and cot t = 3/√7

not clear on your typing for
(cot2t)(sin2t) + sin2 t
is that last part sin^2 t ??????

To simplify trig identities, it is helpful to utilize the fundamental trigonometric identities. Here are explanations for simplifying the given trig identities step by step:

1. (csc t) (sin t):
The reciprocal identity for sine is csc t = 1/sin t. Using this identity, we can rewrite the expression as (1/sin t) * sin t. The sine function cancels out, leaving us with 1.

Therefore, (csc t) (sin t) simplifies to 1.

2. cos t = 0.75 and tan t = 0.88, find sec t and cot t:
From the equation cos t = 0.75, we can use the reciprocal identity for cosine, sec t = 1/cos t, to find sec t. Substituting the given value, we have sec t = 1/0.75, which simplifies to sec t = 4/3.

Next, using the given equation tan t = 0.88, we can use the reciprocal identity for tangent, cot t = 1/tan t, to find cot t. Substituting the given value, we have cot t = 1/0.88, which simplifies to cot t = 25/22.

Therefore, sec t = 4/3 and cot t = 25/22.

3. (cot^2 t) (sin^2 t) + sin^2 t:
First, we can simplify (cot^2 t) (sin^2 t) as (cos^2 t / sin^2 t) * (sin^2 t), since cot t = cos t / sin t. The sin^2 t terms cancel out, leaving us with cos^2 t.

Therefore, (cot^2 t) (sin^2 t) + sin^2 t simplifies to cos^2 t.

4. (csc^2 t - cot^2 t) / sin^2 t:
To simplify this expression, we can use the Pythagorean identities: sin^2 t + cos^2 t = 1, and 1/cos^2 t = tan^2 t + 1.

First, simplify the numerator using the Pythagorean identity: csc^2 t - cot^2 t = (1/sin^2 t) - (cos^2 t/sin^2 t), which simplifies to (1 - cos^2 t)/sin^2 t.

Now, substitute the identity 1 - cos^2 t = sin^2 t into the numerator: (sin^2 t)/sin^2t = 1.

Therefore, (csc^2 t - cot^2 t) / sin^2 t simplifies to 1.