Sum of the Sequence of Trigonometric Functions
The given sequence to find the sum of is: cosx + sin3x + cos5x + sin7x + ...

Explanation:
- The given sequence is an alternating series of cosine and sine functions, with the arguments increasing by 2 each time.
- To find the sum of this sequence, we need to identify a pattern in the terms.

Pattern in the Sequence:
- The terms in the sequence alternate between cosine and sine functions, starting with cosine.
- The coefficients of the sine functions follow the pattern 3, 7, 11, ..., which can be expressed as (4n-1) where n is the position of the term in the sequence.

Summing the Sequence:
- To find the sum of the sequence, we can express each term in terms of a common trigonometric identity.
- Using the identities cos(a)sin(b) = 1/2[sin(a+b) - sin(a-b)] and sin(a)sin(b) = 1/2[cos(a-b) - cos(a+b)], we can simplify the terms.
- After simplifying, we can group the terms into pairs of sine and cosine functions with the same argument, and then sum them up.
- The sum of these pairs will result in a simplified expression for the sum of the sequence.

Conclusion:
- By identifying the pattern in the sequence and using trigonometric identities to simplify the terms, we can find the sum of the given sequence of trigonometric functions.