Question

$if \: tan \: \gamma = \: 3 \div 4 \: then \: 1 - \cos \: gamma \div 1 + \cos \: gamma$
a) 9
b) 1/9
c) 4
d) 1/4 ​

Answer 1

Given:-

$\tan \gamma = \dfrac{3}{4}$

To find:-

$\dfrac{1-\cos \gamma}{1+\cos \gamma} \qquad ...[1]$

Solution:-

We know that,

$\tan\rm \gamma = \dfrac{Opposite \ side}{ Adjacent \ side}$

Hence,

$\tan \rm\gamma = \dfrac{Opposite \ side}{ Adjacent \ side} = \dfrac{3}{4}$

Opposite side = 3 , Adjacent side = 4

Then hypotenuse

(3)² + (4)² = (Hypo)²

→ 9+16 = ( Hypo )²

→ 25 = ( Hypo)²

→ √25 = Hypotenuse

→ 5 = Hypotenuse

We know that

$\cos\rm \gamma = \dfrac{Adjacent \ Side}{Hypotenuse}$

Hence,

$\cos\rm \gamma = \dfrac{Adjacent \ Side}{Hypotenuse}= \dfrac{4}{5}$

Substitute the value of cos γ in [1]

$\dashrightarrow\dfrac{1-\cos \gamma}{1+\cos \gamma}$

$\dashrightarrow\dfrac{1-\frac{4}{5}}{1+\frac{4}{5}}$

$\dashrightarrow\dfrac{\frac{5-4}{5}}{\frac{5+4}{5}}$

$\displaystyle\dashrightarrow\dfrac{\frac{1}{5}}{\frac{9}{5}}$

$\dashrightarrow\dfrac{1}{5} \times \dfrac{5}{9}$

$\bf\dashrightarrow\dfrac{1}{9}$

Hence, option (b) is correct!