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(tan theta + sec theta -1)/(tan theta -sec theta +1) = sec theta +tan theta

(tan theta + sec theta -1)/(tan theta -sec theta +1) = sec theta +tan theta

Grade:10

2 Answers

Arun
25750 Points
6 years ago
Dear Nithesh
 
Numerator = tan theta + sec theta -1 
= sin theta/cos theta + 1/cos theta - 1 
= (sin theta + 1)/cos theta -1 

denominator = (sin theta -1)/cos theta +1 

multiply both by cos theta 

numerator = sin theta +1 - cos theta 
= 2 sin theta/2 cos theta/2 + 2 sin^ theta/2 

denominator = 2 sin theta/2 cos theta/2 - 2 sin ^2 theta/2 

divide 

number = (cos theta/2 + sin theta/2)/(cos theta/2 - sin theta/2) 

= (cos theta/2 + sin theta/2)^2/(cos^2 theta/2- sin ^2 theta/2) 
= (cos^2 theta/2+ sin^2 theta/2 + 2 cos theta/2 sin theta/2)/(cos theta) 
= (1+ sin theta)/ cos theta
= sec theta+ tan theta
 
Hence Proved
ankit singh
askIITians Faculty 614 Points
3 years ago
convert all into sine and cosine then further simplify to get the result
(cos theta/2 + sin theta/2)^2/(cos^2 theta/2- sin ^2 theta/2) 
= (cos^2 theta/2+ sin^2 theta/2 + 2 cos theta/2 sin theta/2)/(cos theta) 
= (1+ sin theta)/ cos theta
= sec theta+ tan theta

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