Verify the identity 1sinx1xsinx1tan2x2tanxsecxsec2Solution1s

Verify the identity.

1/sinx+1/x/sinx-1=tan^2x+2tanxsecx+sec^2

Solution

[(1/sinx)+1]*[(1/sinx)-1]

=[(1+sinx)/sinx]*[(1-sinx)/sinx]

=(1+sinx)/(1-sinx)

multiply and divide by 1+sinx

=[(1+sinx)(1+sinx)]/[(1-sinx)(1+sinx)]

=[1+2sinx+sin²x]/[1-sin²x]

write 1-sin²x =cos²x since cos²x+sin²x=1

=[1+2sinx+sin²x]/[cos²x]

=(1/cos²x)+(2sinx/cos²x)+(sin²x/cos²x)

=sec²x +2(sinx/cosx)(1/cosx) +(tan²x)

=sec²x +2tanxsecx+tan²x

=tan²x +2tanxsecx+sec²x

so [(1/sinx)+1]*[(1/sinx)-1]=tan²x +2tanxsecx+sec²x