Question

निम्नलिखित के मान निकालिए:
(i) $sin60^{o} cos30^{o} + sin30^{o}cos 60^{o}$
(ii) $2tan ^{2} 45^{o} + cos ^{2} 30^{o}- sin ^{2} 60^{o}$
(iii) $\frac{cos 45^{o}}{sec 30^{o}+cosec 30^{o}}$,
(iv) $\frac{sin 30^{o}+tan 45^{o}-cosec 60^{o}}{sec 30^{o}+cos 60^{o}+cot 45^{o}}$,
(v) $\frac{ 5 cos^{2} 60^{o}+4 sec^{2} 30^{o}-tan^{2} 45^{o}}{sin^{2}30^{o}+cos^{2} 30^{o}}$

Answer 1

निम्नलिखित के मान निकालिए:

(i) $sin60^{o} cos30^{o} + sin30^{o}cos 60^{o}$

$\sin(60) \cos(30) + \sin(30) \cos(60) \\ \\ = \frac{ \sqrt{3} }{2} \times \frac{ \sqrt{3} }{2} + \frac{1}{2} \times \frac{1}{2} \\ \\ = \frac{3}{4} + \frac{1}{4} \\ \\ = \frac{4}{4} \\ \\ = 1$


(ii) $2tan ^{2} 45^{o} + cos ^{2} 30^{o}- sin ^{2} 60^{o}$

$2 \times ( {1)}^{2} + ( { \frac{ \sqrt{3} }{2} })^{2} - ({ \frac{ \sqrt{3} }{2} })^{2} \\ \\ = 2$
(iii) $\frac{cos 45^{o}}{sec 30^{o}+cosec 30^{o}}$,

$\frac{ \frac{1}{ \sqrt{2} } }{ \frac{2}{ \sqrt{3} } + 2} \\ \\ = \frac{ \frac{1}{ \sqrt{2} } }{ \frac{2 + 2 \sqrt{3} }{ \sqrt{3} } } \\ \\ = \frac{ \sqrt{3} }{ \sqrt{2}(2 + 2 \sqrt{3} ) } \\ \\ = \frac{ \sqrt{3} }{2 \sqrt{2} (1 + \sqrt{3}) } \times \frac{(1 - \sqrt{3} )}{(1 - \sqrt{3}) } \\ \\ = \frac{ \sqrt{3}(1 - \sqrt{3} )}{2 \sqrt{2} (1 - 3)} \\ \\ = \frac{3 - \sqrt{3} }{4 \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } \\ \\ = \frac{ \sqrt{2}(3 - \sqrt{3}) }{8}$

(iv) $\frac{sin 30^{o}+tan 45^{o}-cosec 60^{o}}{sec 30^{o}+cos 60^{o}+cot 45^{o}}$,

$\frac{ \frac{1}{2} + 1 - \frac{2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } + \frac{1}{2} + 1} \\ \\ = \frac{ \sqrt{3} + 2 \sqrt{3} - 4 }{4 + \sqrt{3} +2 \sqrt{3} } \\ \\ = \frac{3 \sqrt{3} - 4 }{3 \sqrt{3} + 4} \\ \\ = \frac{3 \sqrt{3} - 4 }{3 \sqrt{3} + 4} \times \frac{3 \sqrt{3} - 4 }{3 \sqrt{3} - 4} \\ \\ \\ = \frac{ {(3 \sqrt{3} - 4)}^{2} }{27 - 16} \\ \\ = \frac{27 + 16 - 24 \sqrt{3} }{11} \\ \\ = \frac{43 - 24 \sqrt{3} }{11}$



(v) $\frac{ 5 cos^{2} 60^{o}+4 sec^{2} 30^{o}-tan^{2} 45^{o}}{sin^{2}30^{o}+cos^{2} 30^{o}}$

$\frac{5 \times ( { \frac{1}{2} })^{2} + 4( { \frac{ 2 }{ \sqrt{3} } })^{2} - 1 }{ ( { \frac{1}{2} } )^{2} + ({ \frac{ \sqrt{3} }{2} })^{2} } \\ \\ = \frac{ \frac{5}{4} + \frac{16}{3} - 1 }{ \frac{1}{4} + \frac{3}{4} } \\ \\ = \frac{ \frac{15 + 64 - 12}{12} }{1} \\ \\ = \frac{67}{12}$