Find the derivatives of each function and simplify as much

Question

Part A- Assignment

1. Prove sec²x-1 = sec²x sin²x, and justify significant steps.

2. Prove that sin(x+π) = -sinx.

3. Prove sin⁴θ-cos⁴θ/sin²θ-cos²θ, and justify significant steps.

4. Prove sinx/1+cosx + 1+cosx/sinx = 2cscx, and justify significant steps.

Part B- Assignment: Trigonometric Equations

1. Solve tan²θ - sec θ - 1 = 0.

2. Solve √3 cot3x+1 = 0, where 0 ≤ x ≤ 2π.

Part C - Assignment: Limits of Trigonometric Functions

1. Evaluate the following limits.

a) lim_x→0 x cotx

b) Evaluate the following limit: lim_θ→0 sin(4θ)/5θ

c) Evaluate the following limits: lim_x→0 sinx/tanx

2. If f(x) = sinx, evaluate lim_h→0 (f(2+h) - f(2)/h)

Part D- Derivatives of Trigonometric Functions  

1. Find the derivatives of each function, and simplify as much as possible.

a. y = -3x cosx

b. y = 2csc3(√x)

c. y = (x/2) - (sin2x/4)

d. cos³(5x² - 6)

e. f(x) = sin(5x)/cos(x²)

f. y = 3sin⁴(2-x)^-1

2. Find the derivatives of the following using implicit differentiation.

a. x = siny + cosx

b. xy - y³ = sinx

3. Use a calculator to approximate the slope of the tangent line drawn to the graph of  y = 2sinx + cosx at x = 4.2. Round your answer to two decimal places.

4. Write the equation of the tangent line of y = 2cosx at x = -(π/2).

Part E- Final Module Assignment

1. Find the derivative of the following equation:

y = (sin√(x-1))³

2. Prove the identify tanx+1/tanx - 1 = secx + cscx/sec x - cscx.

3. Find the derivative of y = cos³x sin²2x.              

4. Find the equation of the line tangent to y = 2secx at x = π/4.

5. Find all of the solutions that satisfy sin2x + 3 cosx = 0.

6. Evaluate lim_h→0 (cos(x+h)-cosx/h). Describe the result of this limit.

7. Evaluate limx→π/4 (1-tanx/sinx - cosx).

8. Find dy/dx implicity if x = tanxy.

9. Differentiate the identify sin2x = 2sinx cosx to develop the identify for cos2x, in terms of sinx and cosx.

10. Differentiate the functions y = 1-cox/sinx and y = cscx - cotx, and show that the derivatives are equal.