If x is in Quadrant II and cosx+sinx=1, the value of (𝑠𝑒𝑐x/tanx) + (𝑐𝑠𝑐x/cotx) is
secx/tanx + cscx/cotx
= (1/cosx)/(sinx/cosx) + (1/sinx)/(cosx/sinx)
= 1/sinx + 1/cosx
= (sinx + cosx)/(sinxcosx) <------- correct simplification
= 1/(sincosx) <------- according to your statement,
back to sinx + cosx = 1 <------- not possible in quad II (only solutions would be x =0, x = 90° )
square both sides:
sin^2 x + cos^2 x + 2sinxcosx = 1
1 + 2sinxcosx = 1
sinxcosx = 0
which makes secx/tanx + cscx/cotx = 1/0, which would be undefined
better check your question!