(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
Nithesh , 8 Years ago
Grade 10
Trigonometry> (tan theta + sec theta -1)/(tan theta -se...
2 Answers
Arun
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
= 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
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
= (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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