If m = sec θ - tanθ and n = cosec θ + cot θ, then what is the value of m + n ( m - 1) ?

-1

We are given that :-

m = sec θ - tanθ 

= \(\frac{1 - sinθ}{cosθ}\)

and n = cosec θ + cot θ

= \(\frac{1 + cosθ}{sinθ}\)

Now,

m + n ( m - 1)

= m + nm - n

= \(\frac{1 - sinθ}{cosθ}\) + \(\frac{1 + cosθ}{sinθ}\) × \(\frac{1 - sinθ}{cosθ}\) - \(\frac{1 + cosθ}{sinθ}\)

= \(\frac{sinθ  - sin²θ  + ( 1 + cosθ)× ( 1 - sinθ) - cosθ - cos²θ }{sinθ . cosθ }\) 

= \(\frac{sinθ  - 1 + 1 - sinθ + cosθ - sinθ.cosθ  - cosθ  }{sinθ . cosθ }\)

= \(\frac{  - sinθ.cosθ    }{sinθ . cosθ }\)

= - 1