Question

tan +sec-1/tan-sec+1=1+sin/cos.

Answer 1

to prove (tan+sec-1)/(tan-sec+1)=(1+sin)/cos 

multiply LHS by cos /cos to get 
(sin+1-cos) / (sin-1+cos) 

multiply again by cos/cos to get 
(sin.cos+cos-cos^2) / cos(sin-1+cos) 

= ( cos(1+sin) - (1-sin^2) ) / cos(sin-1+cos)
= ( cos(1+sin) - (1+sin)(1-sin) ) / cos(sin-1+cos) 
= ( (1+sin)(cos-1+sin) ) / cos(sin-1+cos) 
= (1+sin)/cos 
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