Trigonometric Equations

To simplify the given expression , we start with trigonometric identities: .

Using this, the expression becomes:

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Now, simplify each term separately. For the first term, we have:

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And for the second term:

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Combine the two simplified fractions:

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The denominators are different, but observe that . Thus, find a common denominator:

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Express each fraction with this common denominator:

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Simplify and combine:

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Now, simplify: using identity .

Thus, substitute back:

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Re-write in terms of sine and cosine:

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Thus, the expression simplifies to

To find the interval where is increasing, we need to find where the derivative is positive. Let us first differentiate . The derivative of with respect to is: where . Now, differentiate : Thus, the derivative of is: For to be increasing, we need . This requires: which simplifies to: This inequality holds for: Thus, the function is increasing in the interval , which is the correct answer.
Thus, the correct answer is .

We are given the expression: Let’s simplify this step by step.
First, simplify the inner expression: Using the trigonometric identity , the expression becomes: This simplifies further to: because . Now substitute this back into the original expression: Now notice that the expression can be seen as part of the original constant factor. Evaluating numerically, you can see that the result simplifies to:
Thus, the correct answer is .