Question

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Answer 1

Given :

x + y + z = π

⇒ ( x/2 + y/2 + z/2 ) = π/2

Take cot both sides :

⇒ cot ( x/2 + y/2 + z/2 ) = cot π/2

⇒ cot ( x/2 + y/2 + z/2 ) = 0

Use cot ( A + B + C ) = (cot A cot B cot C - cot A - cot B - cot C) / (cot A cot B + cot B cot C + cot A cot C)

[ This is the only formula you are required to remember ]

⇒ (cot x/2 cot y/2 cot z/2 - cot x/2 - cot y/2 - cot z/2 ) / (cot x/2 cot B + cot B cot C + cot A cot C) = 0

Multiplying the denominator with 0 gives 0 :

⇒ ( cot x/2 cot y/2 cot z/2 - cot x/2 - cot y/2 - cot z/2 ) = 0

Transpose - cot x/2 - cot y/2 - cot z/2 and the signs should change :

⇒ cot x/2 + cot y/2 + cot z/2 = cot x/2 cot y/2 cot z/2

Hence it is proved !

Answer 2

Step-by-step explanation:

Given: x + y + z = π.

$=(\frac{x}{2}) + (\frac{y}{2}) + (\frac{z}{2}) = \frac{\pi}{2}$

$=(\frac{x}{2})+(\frac{y}{2})=(\frac{\pi}{2}-\frac{z}{2})$

Apply 'cot' on both sides, we get

$=cot(\frac{x}{2} + \frac{y}{2}) = cot(\frac{\pi}{2} - \frac{z}{2})$

$=\frac{cot(\frac{x}{2}) cot(\frac{y}{2})-1}{cot(\frac{x}{2}) + cot(\frac{y}{2})} = tan \frac{z}{2}$

$=\frac{cot(\frac{x}{2})cot(\frac{y}{2})-1}{cot(\frac{x}{2}) + cot(\frac{y}{2})} = \frac{1}{cot(\frac{z}{2})}$

$=cot(\frac{x}{2})*cot(\frac{y}{2})*cot(\frac{z}{2})-cot(\frac{z}{2}) = cot(\frac{x}{2}) + cot(\frac{y}{2})$

$=cot(\frac{x}{2}) + cot(\frac{y}{2}) + cot(\frac{z}{2}) = cot(\frac{x}{2}) * cot(\frac{y}{2}) * cot(\frac{z}{2})$

Hope it helps!