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If sin alpha sin beta-cos alpha cos beta...

If `sin alpha sin beta-cos alpha cos beta+1=0`, then prove that `1+cotalpha tan beta=0`.

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Given, `sin alpha cos beta-cos alpha cos beta+1=0`
or `cos alpha cos beta-sin alpha sin beta-1`
or `cos (alpha+beta)=1`
Now `1+cot alpha tan beta=1+(cos alpha)/(sin alpha)xx(Sin beta)/(cos beta)`
`=(sin alpha cos beta+cos alpha sin beta)/(sin alpha cos beta)`
`=(sin(alpha+beta))/(sin alpha cos beta)`
`=(0)/(sin alpha cos beta)=0`
`[because sin^(2)(alpha+beta)=1-cos^(2)(alpha+beta)=1-1=0]`
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