Question

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Answer 1

Explanation:

Given expression is,

`\text{cot}A=(1)/(2)(\text{cot}(A)/(2)-\text{tan}(A)/(2))`

To prove this identity we will take the right side of the identity,

`(1)/(2)(\text{cot}(A)/(2)-\text{tan}(A)/(2))=(1)/(2)(\frac{1}{\text{tan}(A)/(2)}-tan(A)/(2))`

`=(1)/(2)(\frac{1-\text{tan}^2(A)/(2)}{tan(A)/(2)})`

`=(1)/(2)[\frac{2(1-\text{tan}^2(A)/(2))}{2tan(A)/(2)}]`

`=(1)/(2)(\frac{2}{\text{tan}A} )` [Since
`\text{tan}A=\frac{2\text{tan}(A)/(2)}{1-\text{tan}^2(A)/(2)}`]

= cot A

Hence right side of the equation is equal to the left side of the equation.