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Let's solve the questions one by one:

**(i) Find the sign of \( \cos 400^\circ \):**
- To find \( \cos 400^\circ \), we first reduce the angle to an equivalent angle within the range \( [0^\circ, 360^\circ] \).
  \[
  400^\circ - 360^\circ = 40^\circ
  \]
  So, \( \cos 400^\circ = \cos 40^\circ \).

  Since \( 40^\circ \) is in the first quadrant, where cosine is positive, the sign of \( \cos 400^\circ \) is **positive**.

**(ii) Express \( 2\cos 35^\circ \cos 75^\circ \) as a sum or difference of two trigonometric functions:**
- We use the product-to-sum identity for cosine:
  \[
  \cos A \cos B = \frac{1}{2} [\cos(A - B) + \cos(A + B)]
  \]
  Applying this identity with \( A = 35^\circ \) and \( B = 75^\circ \):
  \[
  2 \cos 35^\circ \cos 75^\circ = \cos(35^\circ - 75^\circ) + \cos(35^\circ + 75^\circ)
  \]
  Simplifying:
  \[
  = \cos(-40^\circ) + \cos(110^\circ)
  \]
  Since \( \cos(-\theta) = \cos(\theta) \), this becomes:
  \[
  \cos 40^\circ + \cos 110^\circ
  \]
  So, \( 2 \cos 35^\circ \cos 75^\circ = \cos 40^\circ + \cos 110^\circ \).

**(iii) Evaluate \( (i^{131} + i^{49}) \):**
- We know the powers of \( i \) (the imaginary unit) repeat in cycles of 4:
  \[
  i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1
  \]
  To evaluate \( i^{131} \), we divide 131 by 4 and find the remainder:
  \[
  131 \div 4 = 32 \text{ remainder } 3
  \]
  So, \( i^{131} = i^3 = -i \).

  Similarly, for \( i^{49} \):
  \[
  49 \div 4 = 12 \text{ remainder } 1
  \]
  So, \( i^{49} = i^1 = i \).

  Therefore:
  \[
  i^{131} + i^{49} = -i + i = 0
  \]

**(iv) Write the conjugate of the complex number \( \sqrt{5} - i \):**
- The conjugate of a complex number \( a + bi \) is \( a - bi \).
  Therefore, the conjugate of \( \sqrt{5} - i \) is \( \sqrt{5} + i \).

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Would you like further clarification on any part of this solution?

Here are some related questions you might explore:
1. What is the general rule for finding the sign of trigonometric functions in any quadrant?
2. Can product-to-sum identities be extended to sine functions as well?
3. How do powers of imaginary numbers behave for different roots of unity?
4. What other methods exist for simplifying complex expressions with \( i \)?
5. How can you geometrically interpret the conjugate of a complex number?

**Tip:** Always check the quadrant of an angle when finding the sign of trigonometric functions.

Answer the following: (i) Find the signs of the following: cos 400°. (ii) Express the following as a sum or difference of two trigonometric functions: 2cos35°cos75°. (iii) Evaluate (i^131 + i^49). (iv) Write the conjugates of the following complex number.

This detailed solution covers how to find the sign of cos 400°, express 2cos(35°)cos(75°) as a sum of trigonometric functions using identities, evaluate powers of imaginary numbers (i^131 + i^49), and find the conjugate of a complex number. Suitable for students in Grade 10-12.

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