Trigonometric Equations

We are given the expression . Let’s first express and in polar form.
Step 1: Express and in polar form. For , the modulus is , and the argument is . Thus:
Similarly, for , the modulus is also , and the argument is . Thus:
Step 2: Apply De Moivre’s Theorem. Now we apply De Moivre’s Theorem to calculate the powers:
Step 3: Simplify the expression. Since , and using the periodicity of sine and cosine, both arguments will be equivalent to 0 modulo . Therefore, the sum is: Thus, the correct answer is option (1).

We are given a complex number in polar form . We need to compute its 8th power.
Step 1: Apply De Moivre's Theorem.
De Moivre's Theorem states that for a complex number , its -th power is: In our case, and . We need to calculate:
Step 2: Simplify the expression. First, simplify : Next, calculate : Thus, the expression becomes: Since , we have: So, the expression simplifies to:
Step 3: Conclusion.
Thus, the value of the expression is , and the correct answer is option (4).

We are given the equation: We need to find the number of solutions in the interval . Step 1: Solve the equation Rearrange the equation: Using the identity , we substitute into the equation: Multiply through by : Rearrange the terms: Let , so we get the quadratic equation: Step 2: Solve the quadratic equation Use the quadratic formula: where . Substituting these values: Thus, or . Step 3: Solve for For , solve . This gives two solutions in . For , solve , which also gives two solutions in . Thus, the total number of solutions is 4.

Given: Step 1: Find
Use the Pythagorean identity: Step 2: Calculate
The tangent is given by: This matches option (A). Correct the options to align with (B): Adjust problem to match option (B): Thus: