Trigonometry

The value of sin² 51° + sin² 39° is:

We know that .

Therefore, .

So, sin² 51° + sin² 39° = cos² 39° + sin² 39°

Using the trigonometric identity , we get:

cos² 39° + sin² 39° = 1

Answer: (A) 1

The given problem asks for the value of:

We can use the key trigonometric identity:

So, for each pair and , their product is:

Therefore, the product of tangents from to can be paired as:

Each pair gives a product of 1, and since the number of terms is odd (with in the middle), the value of the entire product is: The correct answer is (B) : 1.

The correct answer is: (C): 1

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

Step 1: Simplify the sum of logarithms

The given expression involves a sum of logarithms. Using the logarithmic identity , we can combine all the logarithms into one:

Step 2: Use the symmetry of the tangent function

We know that , so each pair of terms and will multiply to 1:

Thus, the entire product simplifies to 1:

Step 3: Simplify the expression

Now substitute this result back into the original expression:

Since , we have:

Conclusion:
The value of the expression is 1, so the correct answer is (C): 1.

Given: Right-angled triangle with right angle at .

Step 1: Label the sides.

Let:

• (hypotenuse)

Step 2: Express and .

Step 3: Compute .

Step 4: Apply Pythagorean theorem.

Option Analysis:

(A) - Incorrect (sum of sides, not tangent sum)

(B) - Incorrect form

(C) - Correct (matches our derivation)

(D) - Incorrect form

Final Answer:

1. Understand the problem:

Given cos⁻¹x + cos⁻¹y + cos⁻¹z = 3π, we need to find the value of x(y+z) + y(z+x) + z(x+y).

2. Analyze the given equation:

The maximum value of cos⁻¹θ for any θ ∈ [-1,1] is π. The sum equals 3π only if each term equals π.

Thus, cos⁻¹x = cos⁻¹y = cos⁻¹z = π ⇒ x = y = z = -1.

3. Substitute x = y = z = -1:

The expression becomes:

(-1)(-1-1) + (-1)(-1-1) + (-1)(-1-1) = (-1)(-2) + (-1)(-2) + (-1)(-2) = 2 + 2 + 2 = 6

Correct Answer: (C) 6

From the given condition: Using the identity , we can express as: Substitute this into the original equation: Simplify: Combine like terms: Solve for : Now, find : From the earlier equation: Substitute back: Given ,

However, the simplified expression for is when evaluated at the specific solution.

Hence, the correct answer is .

1. Understand the problem:

We are given two composite functions:

We need to determine the functions and from the given options.

2. Analyze the first condition :

This means .

If , then , which suggests (where ).

3. Verify with the second condition :

If and , then:

This matches the given condition.

Correct Answer: (C)