Trigonometric Identities

Given equations:

Using sum-to-product identities:

Substituting given values:

Dividing both equations:

So, , which gives:

, thus .

Solving for :

Squaring both sides and using :

Thus, the correct answer is:

We are given the expression: We can expand each square term: Now, adding the two expanded expressions: Using the Pythagorean identity , we simplify: Using the sum formula for cosines, , we get: Since in a right triangle, . Therefore, the final expression becomes:

The correct option is (B) :

Given:

• Real numbers with
• Equation:

Step 1: Rewrite the equation using trigonometric identity: where is an angle such that and .

Step 2: Analyze the range of the left side:

Step 3: Compare with the given condition :

• If , the equation has no solution since the maximum value of cosine is 1.
• Similarly, if , there's no solution since the minimum value of cosine is -1.

Conclusion: The equation has no real solution under the given condition.

The correct answer is (D) no real solution.

Step 1: Let , , . Then:

Step 2: Using the trigonometric identity for angles summing to :

Step 3: Substitute back the original variables:

The correct answer is (B) 1.

To solve for in the equation:

we first need to acknowledge the domain restrictions of the inverse trigonometric functions. The range of is , and the valid domain for within is . Thus, for the expression to be valid, it must satisfy:

Dividing the entire inequality by 2 gives:

Next, we examine the given equation:

This implies the angle radians must correspond to a valid cosine value that 2 times the sine of can achieve. However, converting radians to degrees or using common knowledge of trigonometric values shows that it falls outside typical ranges used for trigonometric tables, and must be evaluated correctly in radians.

The cosine of an angle outside the domain limits means the equation has no valid solution under real number constraints. Therefore, none of the offered options, each incorrectly implying valid output for based on given inputs, match possible outcomes.

Thus, the correct conclusion is: None of these

We are given: Using the identity for the sum of inverse cotangents: First, compute : Now, add : Thus: So the answer is .