You've uploaded an image showing trigonometric equations labeled as (h), (i), (j), and (k). Let me analyze and provide an explanation for each equation.

(h)

$\cos A + \cos B + \cos C = 1 + 4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$ This identity is a trigonometric relationship, often used in triangle trigonometry, expressing the sum of cosines of angles in terms of their halves and the product of sines.

1. Left-hand side (LHS):
   Using the sum-to-product formulas:
   $\cos A + \cos B = 2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}$ Substituting $A + B + C = 180^\circ \Rightarrow C = 180^\circ - (A + B)$, the equation simplifies.

2. Right-hand side (RHS):
   The term $4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$ represents a product form of half-angles. Verifications using known identities will show both sides match.

(i)

$\sin 2A + \sin 2B + \sin 2C = 4 \sin A \sin B \sin C$ This equation is derived from the sum-to-product identities: $\sin x + \sin y = 2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}$ By expressing $2A + 2B + 2C = 360^\circ$, we substitute and simplify. The result transforms into a product of sines.

(j)

$\sin \frac{A}{2} - \cos \frac{B - C}{2} = -2 \sin \frac{B}{2} \sin \frac{C}{2}$ This involves reducing both LHS and RHS using: $\sin x - \cos y = \sin(x + y) - \sin(x - y)$ Further expansion of $A, B, C$ in terms of triangle properties validates the identity.

(k)

$1 + \cos 2C - \cos 2A - \cos 2B = 4 \sin A \sin B \cos C$ Using double-angle and product-to-sum formulas: $\cos 2x = 1 - 2\sin^2 x$ Simplify and verify both sides equal.

Would you like step-by-step derivations for any of these?

Related Questions:

1. How are sum-to-product identities applied to simplify trigonometric expressions?
2. What are the relationships between the angles in a triangle when using these formulas?
3. How does the identity $\sin 2x = 2\sin x\cos x$ aid in verifying (i) and (k)?
4. What role do half-angle formulas play in solving equation (h)?
5. How can these identities be used in solving geometric problems involving triangles?

Tip:

Always verify trigonometric identities using known properties of angles in a triangle, such as $A + B + C = 180^\circ$.