Question

prove this identities... ​

Answer 1

Answer:

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Answer 2

First consider LHS...

sin∅÷(cot∅+cosec∅)

=sin∅÷(cos∅/sin∅ + 1/sin∅)

=sin∅÷((cos∅+1)/sin∅)

=sin²∅÷(cos∅+1)

=(1 - cos²∅)÷(cos∅+1)

=(1 - cos∅)(1 + cos∅) ÷ (cos∅+1)

= 1 - cos∅ { After cancellation}

Now, let's consider RHS

2 + sin∅÷(cot∅ - cosec∅)

{By Rationalising with (cot∅+cosec∅) and with the help of the identity 1+cot²∅= cosec ²∅, we get}

= 2 - sin∅(cot∅+cosec∅)

= 2 - sin∅{(cos∅/sin∅)+(1/sin∅)}

= 2 - sin∅{(cos∅+1)/sin∅}

Sin∅ would be cancelled...

= 2 - (cos∅+1)

= 2 - cos∅ - 1

= 1 - cos∅

As LHS=RHS,

Hence proved.