Question

If $\rm{x +y + z = \pi}$prove the trigonometric identity $\tt\blue{\cot\frac{x}{2 } + \cot\frac{y}{2} + \cot\frac{z}{2} = \cot \frac{x}{2} \cot \frac{y}{2} \cot \frac{z}{2}}$​

Answer 1

Refer the solution in attachment.

Some useful formulae:

• cos (A + B) = (cos A•cos B - 1)/(cos A + cos B)
• cos (A - B) = (cos A•cos B + 1)/(cos A - cos B)
• sin (A + B) = sin A•cos B + cos A•sin B
• sin (A - B) = sin A•cos B - cos A•sin B
• cos (A + B) = cos A•cos B - sin A•sinB
• cos (A - B) = cos A•cos B + sin A•sinB
• tan (A + B) = (tan A + tan B)/(1 - tan A•tan B)
• tan (A + B) = (tan A - tan B)/(1 + tan A•tan B)
• sin 2A = 2 sin A• cos A
• sin 2A = (2 tan A)/(1 + tan²A)
• cos 2A = cos²A - sin²A
• cos 2A = 1 - 2 sin²A
• cos 2A = 2 cos²A - 1
• cos 2A = (1 - tan²A)/(1 + tan²A)
• tan 2A = (2 tan A)/(1 - tan²A)
• sin 3A = 3 sin A - 4 sin³A
• cos 3A = 4 cos³A - 3 cos A
• tan 3A = (3 tan A - tan³A)/(1 - 3 tan²A)
• cot 3A = (cot³A - 3 cot A)/(3 cot²A - 1)

Answer 2

Given ,

$x + y + z = \pi \\ x + y = \pi - z \\ multiplying \: \frac{1}{2} \: both \: sides \\ \\ \frac{x + y}{2} = \frac{\pi - z}{2} \\ \\ \frac{x}{2} + \frac{y}{2} = \frac{\pi}{2} - \frac{z}{2} \\ \\ taking \: cot \: both \: sides \\ \\ cot( \frac{x}{2} + \frac{y}{2} ) = cot( \frac{\pi}{2} - \frac{z}{2} ) \\ \\ \frac{cot \frac{x}{2} \times cot \frac{y}{2} - 1}{cot \frac{x}{2} + cot \frac{y}{2} } = \frac{cot \frac{\pi}{2} \times cot \frac{z}{2} + 1}{cot \frac{z}{2} - cot \frac{\pi}{2} } \\ \\ \frac{cot \frac{x}{2} \times cot \frac{y}{2} - 1}{cot \frac{x}{2} + cot \frac{y}{2} } = \frac{0 + 1}{cot \frac{\pi}{2} - 0 } \\ \\ \frac{cot \frac{x}{2} \times cot \frac{y}{2} - 1 }{cot \frac{x}{2} + cot \frac{y}{2} } = \frac{1}{cot \frac{z}{2} } \\ \\ cross \: multiplication \\ \\ cot\frac{z}{2} (cot \frac{x}{2} \times cot \frac{y}{2} - 1) = cot \frac{x}{2} + cot \frac{y}{2} \\ \\ cot \frac{x}{2} \times cot \frac{y}{2} \times cot \frac{z}{2} - cot \frac{z}{2} = cot \frac{x}{2} + cot \frac{y}{2} \\ \\ cot \frac{x}{2} \times cot \frac{y}{2} \times cot \frac{z}{2} = cot \frac{x}{2} + cot \frac{y}{2} + cot \frac{z}{2} .$

hence , L.H.S. = R.H.S. (proved) !!

hopefully its helped u dear :)