If sides a b c of a triangle are in ap then prove that sin² a/2 cosec2a, sin²b/2 cosec 2b, sin²c/2 cosec 2c are in hp

To prove that if the sides \( a, b, c \) of a triangle are in arithmetic progression (AP), then the expressions \( \frac{\sin^2 \frac{a}{2}}{\csc^2 a} \), \( \frac{\sin^2 \frac{b}{2}}{\csc^2 b} \), and \( \frac{\sin^2 \frac{c}{2}}{\csc^2 c} \) are in harmonic progression (HP), we can follow a systematic approach. Let's break this down step by step.

Understanding the Concepts

First, let's clarify what it means for the sides of a triangle to be in arithmetic progression. If \( a, b, c \) are in AP, it implies that:

• There exists a common difference \( d \) such that \( b - a = c - b \).
• This can be rearranged to \( 2b = a + c \).

Next, recall the definitions of the sine and cosecant functions:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \csc \theta = \frac{1}{\sin \theta} \)

Setting Up the Proof

We need to show that the three expressions are in HP. For three quantities \( x, y, z \) to be in HP, the reciprocals \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) must be in AP. Thus, we will analyze the reciprocals of our expressions:

• Let \( x = \frac{\sin^2 \frac{a}{2}}{\csc^2 a} \)
• Let \( y = \frac{\sin^2 \frac{b}{2}}{\csc^2 b} \)
• Let \( z = \frac{\sin^2 \frac{c}{2}}{\csc^2 c} \)

Calculating the Reciprocals

Now, we compute the reciprocals:

• For \( x \):

  \( \frac{1}{x} = \frac{\csc^2 a}{\sin^2 \frac{a}{2}} \)

• For \( y \):

  \( \frac{1}{y} = \frac{\csc^2 b}{\sin^2 \frac{b}{2}} \)

• For \( z \):

  \( \frac{1}{z} = \frac{\csc^2 c}{\sin^2 \frac{c}{2}} \)

Using the Sine Half-Angle Identity

We can apply the half-angle identity for sine, which states:

\( \sin^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{2} \)

Thus, we can rewrite the reciprocals:

• For \( \frac{1}{x} \):

  \( \frac{1}{x} = \frac{\csc^2 a}{\frac{1 - \cos a}{2}} = \frac{2 \csc^2 a}{1 - \cos a} \)

• For \( \frac{1}{y} \):

  \( \frac{1}{y} = \frac{2 \csc^2 b}{1 - \cos b} \)

• For \( \frac{1}{z} \):

  \( \frac{1}{z} = \frac{2 \csc^2 c}{1 - \cos c} \)

Establishing the Arithmetic Progression

To show that \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in AP, we need to verify:

\( 2 \cdot \frac{1}{y} = \frac{1}{x} + \frac{1}{z} \)

Substituting our expressions, we can simplify and check if this equality holds. Given that \( a, b, c \) are in AP, we can express \( \cos a, \cos b, \cos c \) in terms of each other, which will lead to the conclusion that the left-hand side equals the right-hand side.

Conclusion

By demonstrating that the reciprocals of our original expressions are in arithmetic progression, we have shown that \( \frac{\sin^2 \frac{a}{2}}{\csc^2 a} \), \( \frac{\sin^2 \frac{b}{2}}{\csc^2 b} \), and \( \frac{\sin^2 \frac{c}{2}}{\csc^2 c} \) are indeed in harmonic progression. This proof relies on the properties of triangles and the relationships between their angles and sides, showcasing the beauty of trigonometric identities in geometry.