Given that cos z = L , whrere z is an acute angle, find an expression for (frac{cot z – csc z}{sec z + tan z})
-
A.
(frac{1 – L}{1 + L}) -
B.
(frac{L^2 sqrt{3}}{1 + L}) -
C.
(frac{1 + L^3}{L^2}) -
D.
(frac{L(L – 1)}{1 – L + 1 sqrt{1 – L^2}})
Correct Answer: Option D
Explanation
Given Cos z = L, z is an acute angle
(frac{text{cot z – cosec z}}{text{sec z + tan z}}) = cos z
= (frac{text{cos z}}{text{sin z}})
cosec z = (frac{1}{text{sin z}})
cot z – cosec z = (frac{text{cos z}}{text{sin z}}) – (frac{1}{text{sin z}})
cot z – cosec z = (frac{L – 1}{text{sin z}})
sec z = (frac{1}{text{cos z}})
tan z = (frac{text{sin z}}{text{cos z}})
sec z = (frac{1}{text{cos z}}) + (frac{text{sin z}}{text{cos z}})
= (frac{1}{l}) + (frac{text{sin z}}{L})
the original eqn. becomes
(frac{text{cot z – cosec z}}{text{sec z + tan z}}) = (frac{L – frac{1}{text{sin z}}}{1 + sin frac{z}{L}})
= (frac{L(L – 1)}{text{sin z}(1 + text{sin z})})
= (frac{L(L – 1)}{text{sin z} + 1 – cos^2 z})
= sin z + 1
= 1 + (sqrt{1 – L^2})
= (frac{L(L – 1)}{1 – L + 1 sqrt{1 – L^2}})