Question

Prove (tan theta+sec theta-1)/tan theta-sec theta +1 =(1+sin theta)/cos theta​

Answer 1

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Here, i am holding the "THETAS" as "A".

So, we can proceed our solution as under :

Taking L.H.S.

L.H.S =

(tan A + sec A - 1)/(tan A - sec A + 1)

= (tan A + sec A - sec²A + tan²A)/(tan A - sec A + 1)

= [tan A + sec A - {(sec A+tan A) (sec A - tan A)}]/[tan A - sec A + 1]

= [tan A + sec A (1 - sec A + tan A)]/(tan A - sec A + 1)

= tan A + sec A

= sin A/cos A + 1/cos A

= ( 1 + sin A ) / cos A

= {R. H. S.}

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ULTIMATELY,

L.H.S. = R.H.S. (PROVED)

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Answer 2

Answer:

Here, i am holding the "THETAS" as "A".

So, we can proceed our solution as under :

Taking L.H.S.

L.H.S =

(tan A + sec A - 1)/(tan A - sec A + 1)

= (tan A + sec A - sec²A + tan²A)/(tan A - sec A + 1)

= [tan A + sec A - {(sec A+tan A) (sec A - tan A)}]/[tan A - sec A + 1]

= [tan A + sec A (1 - sec A + tan A)]/(tan A - sec A + 1)

= tan A + sec A

= sin A/cos A + 1/cos A

= ( 1 + sin A ) / cos A

= {R. H. S.}

==================================

ULTIMATELY,

L.H.S. = R.H.S. (PROVED)