Question

evaluate the value :-

$\frac{ \cos(45) }{ \sec(30) + \csc(30) }$
​

Answer 1

SOLUTION :

❒ To evaluate :-

• $\: \: \bigstar \: \: \: \: \bold{ \dfrac{Cos(45) }{Sec(30) + Csc(30) }}\\ \\ \underline{ \large{ \rm{Or,}}} \\ \\ \: \: \bigstar \: \: \: \: \bold{ \dfrac{Cos \: 45 \degree }{Sec \: 30 \degree + Csc \: 30 \degree }}$

❒ Important Values :-

• $\bigstar \: \: \rm{Cos \: 45 \degree = \cfrac{1}{ \sqrt{2}}}$

• $\bigstar \: \: \rm{Sec \: 30 \degree = \cfrac{2}{ \sqrt{3}}}$

• $\bigstar \: \: \rm{Csc \: 30 \degree = 2}$

❒ Evaluation :-

• ➮ By putting the Value, we get,

$\bigstar \: \: \: \: \sf{\dfrac{Cos \: 45 \degree }{Sec \: 30 \degree + Csc \: 30 \degree }} \\ \\ \sf{= \dfrac{\cfrac{1}{ \sqrt{2}}}{ \cfrac{2}{ \sqrt{3}} + 2}} \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{= \dfrac{\cfrac{1}{ \sqrt{2}}}{ \cfrac{2 + 2 \sqrt{3}}{ \sqrt{3}}}} \: \: \: \: \: \: \: \\ \\ \sf{= \dfrac{\cfrac{1}{ \sqrt{2}}}{ \cfrac{2 \: ( \: 1 + \sqrt{3} \: )}{ \sqrt{3}}}} \: \: \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{= \dfrac{1}{ \sqrt{2} } \times \cfrac{ \sqrt{3} }{2(1 + \sqrt{3} )}} \: \: \: \: \: \: \: \: \\ \\ \sf{ = \dfrac{ \sqrt{3} }{2 \sqrt{2} \: (1 + \sqrt{3} )}} \\ \\ \: \: \: \: \: = \large{ \sf{\dfrac{ \sqrt{3} }{2 \sqrt{2} \: ( \sqrt{3} + 1)}}}$

• ➮ By rationalizing the terms, we get,

$\bigstar \: \: \sf{ \dfrac{ \sqrt{3} }{2 \sqrt{2} \: ( \sqrt{3} + 1)}} \: \: \: \: \: \: \: \: \: \: \: \\ \\ \: \: \: \: \: \: \: \sf{= \dfrac{ \sqrt{3} }{2 \sqrt{2} \: ( \sqrt{3} + 1)}} \times \dfrac{ \sqrt{3} - 1}{ \sqrt{3} - 1} \\ \\ \: \: \: \: \: \sf{ = \dfrac{ \sqrt{3} \: \: ( \sqrt{3} - 1)}{2 \sqrt{2} \: ( \sqrt{3} + 1) \: ( \sqrt{3} - 1)} } \\ \\ \sf{= \dfrac{3 - \sqrt{3} }{2 \sqrt{2} \: (3 - 1) }} \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{= \dfrac{3 - \sqrt{3} }{2 \sqrt{2} \times 2}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{ = \large{ \dfrac{3 - \sqrt{3} }{4 \sqrt{2}}}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

• ➮ By multiply both the numerator and denominator by √2, we get,

$\: \: \bigstar \: \: \: \: \sf{\dfrac{3 - \sqrt{3} }{4 \sqrt{2}}} \\ \\ \sf{ = \dfrac{3 - \sqrt{3} }{4 \sqrt{2}} \times \dfrac{ \sqrt{2} }{ \sqrt{2} } } \: \: \: \\ \\ \sf{ = \dfrac{(3 - \sqrt{3} ) \times \sqrt{2}}{4 \sqrt{2} \times \sqrt{2} }} \: \: \\ \\ \: \: \: \: \sf{ = \dfrac{3 \sqrt{2} - (\sqrt{3} \times \sqrt{2} ) }{4 \times 2} } \\ \\ =\large{ \boxed{ \sf{ \: \dfrac{3 \sqrt{2} - \sqrt{6} }{8} } \: }}$

$\therefore \: \: \boxed{ \rm{ \: \: \dfrac{Cos \: (45) }{Sec \: (30) + Csc \: (30) } = \dfrac{3 \sqrt{2} - \sqrt{6}}{8} \: \: }}$

Answer 2

Hi there!

Plz check the attachment for answer..