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(tanx+secx−1) / (tanx−secx+1)
Multiplying numerator and denominator by tanx+secx+1
= {(tanx+secx−1) / (tanx−secx+1)} × {(tanx+secx+1) / (tanx+secx+1)}
= {tan2 x+tanxsecx+tanx+tanxsecx+sec2 x+secx−tanx−secx−1}/
{tan2 x+tanxsecx+tanx−tanxsecx−sec2 x−secx+tanx+secx+1}
= (tan2 x+2tanxsecx+sec2 x−1) / (tan2 x+2tanx−sec2 x+1)
As sec2x=tan2 x+1, above is equal to
= (tan2 x+2tanxsecx+tan2 x+1−1) / (tan2 x+2tanx−tan2 x−1+1)
= (2tan2 x+2tanxsecx) / 2tanx
= 2tanx(tanx+secx) / 2tanx
= tanx+secx
Now differentiating above wrt x
d/dx(tanx) + d/dx(secx)
=> sec (secx+tanx)
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