If m = sec θ - tanθ and n = cosec θ + cot θ, then what is the value of m + n ( m - 1) ?
Answer & explanation
Correct answer: option 4
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