Question

please tell me how to prove it and also how to prove other questuons of this type

Answer 1

Here is your answer :

R.H.S = ( 1 + cos A ) / sin A

L.H.S = ( cot A + cosec A - 1 ) / ( cot A - cosec A + 1 )

Using identity :

=> cosec² A - cot² A = 1

•°• -1 = cot²A - cosec²A

= [ ( cot A + cosec A ) + ( cot²A - cosec² A ) ] / ( cot A - cosec A + 1 )

Using identity :

[ a² - b² = ( a + b ) ( a - b ) ]

= [ ( cot A + cosec A ) + ( cot A + cosec A ) ( cot A - cosec A ) ] / ( cot A - cosec A + 1 )

Taking ( cot A + cosec A ) common in numerator,

= [ ( cot A + cosec A ) ( 1 + cot A - cosec A ) ] / ( cot A - cosec A + 1 ) ]

= [ ( cot A + cosec A ) ( cot A - cosec A + 1 ) ] / ( cot A - cosec A + 1 )

= ( cot A + cosec A )

Using identity :

=> cot A = cos A / sin A

=> cosec A = 1/sin A

= [ ( cos A / sin A ) + ( 1 / sin A ) ]

= ( cos A + 1 ) / sin A

=> ( 1 + cos A ) / sin A = R.H.S

Proved.

There is not only a type of trigonometry ratios, there is a lot of types in trigonometry .

I would suggest you to solve example questions, derivation of identities etc.

Hope it helps !!