1. Establish the identity:

sin(theta+3pi/2)=-cos(theta)

2. Find the exact value of 2(theta) if sin(theta)=5/13

3. Show that: csc2(theta)-cot2(theta)=tan(theta)

4. Find the exact value of tan(cos^-1(square root of 3/2)

5. Approximate the value rounded to two decimal places: csc^-1(4)

1. To establish the identity sin(theta + 3pi/2) = -cos(theta), we can use the trigonometric angle addition formula. According to the formula, sin(a + b) = sin(a)cos(b) + cos(a)sin(b).

We can rewrite the given expression as sin(theta)cos(3pi/2) + cos(theta)sin(3pi/2). Now, the values of sin(3pi/2) and cos(3pi/2) are well known. sin(3pi/2) = -1 and cos(3pi/2) = 0.

By substituting these values, the expression becomes -cos(theta), which means the identity holds true.

2. To find the exact value of 2(theta) if sin(theta) = 5/13, we need to use the double-angle formula for sin(2theta). The double-angle formula states that sin(2theta) = 2sin(theta)cos(theta).

We know that sin(theta) = 5/13, and we can use the Pythagorean identity sin^2(theta) + cos^2(theta) = 1 to find cos(theta). In this case, cos(theta) = sqrt(1 - sin^2(theta)) = sqrt(1 - (5/13)^2) = 12/13.

Now, substituting these values into the double-angle formula, we have sin(2theta) = 2 * (5/13) * (12/13) = 120/169.

Hence, 2(theta) = sin^(-1)(120/169).

3. To show that csc^2(theta) - cot^2(theta) = tan(theta), we can start with the basic trigonometric definitions.

csc^2(theta) = 1/sin^2(theta) and cot^2(theta) = cos^2(theta)/sin^2(theta).

Substituting these definitions into the left-hand side of the equation, we get 1/sin^2(theta) - cos^2(theta)/sin^2(theta), which could be represented as (1 - cos^2(theta))/sin^2(theta).

Now, we can use the Pythagorean identity sin^2(theta) + cos^2(theta) = 1 to replace (1 - cos^2(theta)). This simplifies the expression to 1/sin^2(theta), which is equal to csc^2(theta).

Therefore, csc^2(theta) - cot^2(theta) simplifies to csc^2(theta) - cos^2(theta)/sin^2(theta) = csc^2(theta) - 1 = 1/sin^2(theta) - 1 = 1 - sin^2(theta)/sin^2(theta) = 1 - 1 = 0.

The right-hand side of the equation is tan(theta). Since the left-hand side simplifies to 0 and the right-hand side is also 0, we have shown that csc^2(theta) - cot^2(theta) = tan(theta).

4. To find the exact value of tan(cos^(-1)(sqrt(3/2))), we need to understand the reference angle and the definition of the inverse cosine function.

The inverse cosine (cos^(-1)) function gives the angle whose cosine is a given value. In this case, the given value is sqrt(3/2). The range of the inverse cosine function is restricted to [0, pi].

To find the reference angle, we use the Pythagorean identity sin^2(theta) + cos^2(theta) = 1. Since cos(theta) = sqrt(3/2), we can find sin(theta) by substituting the value of cos(theta) into the identity: sin(theta) = sqrt(1 - cos^2(theta)) = sqrt(1 - (3/2)) = sqrt(1/2) = sqrt(2)/2.

Now, we have sin(theta) and cos(theta). Since tan(theta) = sin(theta)/cos(theta), we can find tan(theta) by dividing sin(theta) by cos(theta): tan(theta) = (sqrt(2)/2) / (sqrt(3)/2) = sqrt(2/3).

Therefore, tan(cos^(-1)(sqrt(3/2))) = sqrt(2/3).

5. To approximate the value rounded to two decimal places for csc^(-1)(4), we first need to understand the inverse cosecant function (csc^(-1)).

The inverse cosecant function gives the angle whose cosecant is a given value. In this case, the given value is 4. The range of the inverse cosecant function is restricted to [-pi/2, pi/2] excluding zero.

To find the approximate value, we can input 4 into the calculator's inverse cosecant function or use the reciprocal identity csc^(-1)(x) = arcsin(1/x). Using the reciprocal identity, we have csc^(-1)(4) = arcsin(1/4).

Approximating arcsin(1/4) with a calculator or table of values will give you the approximate value rounded to two decimal places.