Q1:

Prove cos^2t+4cost+4/cost+2=2sect+1/sect

Q2:
A ship leaves port with a bearing of S 40 W. After traveling 7 miles, the ship turns 90 degrees on a bearing of N 50 W for 11 miles. At that time, what is the bearing of the ship from port?

Q1: To prove the equation cos^2t + 4cost + 4 / cost + 2 = 2sect + 1 / sect, we can simplify both sides of the equation and show that they are equal.

Step 1: Simplify the left side of the equation
cos^2t + 4cost + 4 / cost + 2

To simplify this expression, we first notice that the numerator can be factored as (cost + 2)^2. Rewriting the equation, we have:

(cos^2t + 4cost + 4) / (cost + 2)

Now, we can factor the numerator further as (cost + 2)^2 = (cos t + 2)^2. The equation becomes:

((cost + 2)^2) / (cost + 2)

Simplifying further, we can cancel out the common factor (cost + 2) in the numerator and denominator:

cost + 2

Step 2: Simplify the right side of the equation
2sect + 1 / sect

To simplify this expression, we can multiply both the numerator and denominator of the fraction by sect:

(2sect + 1) * sect / sect

This simplifies to:

2sect^2 + sect / sect

Canceling out the common factor sect in the numerator and denominator, we get:

2sect + 1

Step 3: Compare the simplified expressions
After simplifying the left and right sides of the equation, we have:

cos^2t + 4cost + 4 / cost + 2 = 2sect + 1

We can see that both sides are equal, so we have proven the equation.

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Q2: To determine the bearing of the ship from the port after it has travelled 7 miles S 40 W, and then turned 90 degrees on a bearing of N 50 W for 11 miles, we can use trigonometry and vector addition.

Step 1: Determine the initial position vector of the ship
The bearing S 40 W can be represented as a vector with components (-7sin40, -7cos40) miles, where sin40 is the sine of 40 degrees and cos40 is the cosine of 40 degrees.

So, the initial position vector is (-7sin40, -7cos40).

Step 2: Determine the displacement vector after the turn
The bearing N 50 W can be represented as a vector with components (-11sin40+11cos50, 11cos40+11sin50) miles.

So, the displacement vector after the turn is (-11sin40+11cos50, 11cos40+11sin50).

Step 3: Add the initial position vector and the displacement vector
To find the final position vector (i.e., the position of the ship from the port), we add the initial position vector and the displacement vector:

Final position vector = (-7sin40, -7cos40) + (-11sin40+11cos50, 11cos40+11sin50)

Simplifying the addition, we get:

Final position vector = (-7sin40-11sin40+11cos50, -7cos40+11cos40+11sin50)

Final position vector = (-18sin40+11cos50, 4cos40+11sin50)

Step 4: Calculate the angle of the final position vector
To determine the bearing of the ship from the port, we need to calculate the angle made by the final position vector with the positive y-axis (north).

Using trigonometry, we can calculate the angle θ as:

θ = arctan((4cos40+11sin50) / (-18sin40+11cos50))

Calculating this angle will give us the bearing of the ship from the port.