Question

$\frac{cos \: 45}{sec \: 30 \: + \: cosec \: 30 }$
​

Answer 1

Answer:

$\frac{ \cos45 \degree}{ \sec30 \degree + \cosec30 \degree } \\ \\ = \frac{ \frac{1}{ \sqrt{2} } }{ \frac{2}{ \sqrt{3} } + 2 } \\ \\ = \frac{ \frac{1}{ \sqrt{2} } }{ \frac{2 + 2 \sqrt{3} }{ \sqrt{3} } } \\ \\ = \frac{ \sqrt{3} }{ \sqrt{2} (2 + 2 \sqrt{3}) } \\ \\ = \frac{ \sqrt{3}(2 - 2 \sqrt{3} )}{ \sqrt{2}( {2}^{2} - {(2 \sqrt{3}) }^{2} } \\ \\ = \frac{ \sqrt{3} (2 - 2 \sqrt{3}) }{ \sqrt{2}(4 - 4 \times 3) } \\ \\ = \frac{ \sqrt{6} (2 - 2 \sqrt{3}) }{2(4 - 12)} \\ \\ = \frac{ \sqrt{6}(2 - 2 \sqrt{3)} }{-16}$

= √6(√3-1)/8

Answer 2

Question:

find the value of

$\dfrac{ \cos(45°) }{ \sec(30°) + \csc(30°) }$

Used formulas:

cos45° = 1/√2

sec30° = 2/√3

cosec30° = 2

Answer:

$= \dfrac{ \frac{1}{ \sqrt{2} } }{ \frac{2}{ \sqrt{3} } + 2 } \\ \\ = \dfrac{ \frac{1}{ \sqrt{2} } }{ \frac{2 + 2 \sqrt{3} }{ \sqrt{3} } } \\ \\ = \dfrac{ \sqrt{3} }{(2 + 2 \sqrt{3} ) \sqrt{2} } \\ \\ = \dfrac{ \sqrt{3} }{2 \sqrt{2} + 2 \sqrt{6} } \\ \\ = \frac{ \sqrt{3} }{2 \sqrt{2} (1 + \sqrt{3} )} \:$

Used concepts:

→ trigonometric ratios

→ trigonometric ratios for standard angles