Trigonometric Identities

Given tan A + cot A = 2. Since , we can rewrite the given equation as:

Let .

Then .

Multiplying by gives , or .

This factors as , so . Thus, , which implies .

Now, tan⁴ A + cot⁴ A = 1⁴ + 1⁴ = 1 + 1 = 2.

Answer: (A) 2

We need to find the value of .

First, note that the domain of both and is . However, . Therefore, and are not defined.

Since and do not exist, the entire expression does not exist.

Thus, the correct option is (D) Does not exist.

Given the function:

To find the derivative, we use the chain rule and the power rule:

Simplifying further:

Now, to determine the intervals of increasing or decreasing, we need to analyze the sign of .

Let's analyze the sign of in different intervals:

In the interval , for values between and , is positive, and is negative.

Since and are non-negative, while is positive, all terms in are positive.

Therefore, is always positive in this interval. Thus, is strictly increasing in the interval .

Based on the analysis, we can conclude that the function

is strictly increasing in the interval , which corresponds to option (D) decreasing in .

The correct answer is: (D): 0

We are tasked with finding the value of the following determinant:

Step 1: Simplify the trigonometric values

First, recall that , because the tangent of 135° is -1 (since ). Therefore, we can replace all instances of in the matrix with -1:

Step 2: Analyze the structure of the matrix

Now, observe that this is a symmetric matrix where the elements in the first and second rows are switched between the columns. This symmetry indicates that the rows are linearly dependent, meaning one row can be expressed as a linear combination of the others.

Step 3: Conclusion

The determinant of a matrix with linearly dependent rows is 0. Therefore, the value of the determinant is 0.

Conclusion:
The value of the determinant is 0, so the correct answer is (D): 0.

The correct answer is: (A):

We are tasked with evaluating the integral:

Step 1: Simplify the expression

The given expression contains both sine and cosine terms. We begin by rewriting the denominator to combine like terms:

After simplifying, we get:

Step 2: Use a substitution to simplify further

Next, we use the substitution , which implies that . Since , we can rewrite the denominator in terms of . The integral becomes:

Step 3: Simplify the expression

After simplifying the denominator, we get a rational function of , which can be integrated using standard methods. The resulting integral simplifies to:

Step 4: Substitute back to original variable

Finally, substitute back into the expression to get the final result:

Conclusion:
The correct answer is (A):

The correct answer is: (D):

We are tasked with evaluating the integral:

Step 1: Simplify the integrand

The given expression involves trigonometric functions , , and . Begin by simplifying the integrand using trigonometric identities. Recall that:

Substituting these identities into the integrand, we get:

Step 2: Apply the integral

Now the integral simplifies to:

Step 3: Use substitution

Let’s use the substitution , so that . Also, when , , and when , again. This converts the integral into:

To proceed with the integral, notice that we have the limits of integration in terms of , and now the integrand involves a simple polynomial in terms of . Evaluate this integral step by step, yielding:

Conclusion:
The correct answer is (D):