We know that,
Sin30° = 1/2
Tan45° = 1
Cosec60° = 2/√3
Sec30°2/√3
Cos60° = 1/2
Cot45° = 1
Given us,
Sin30° + Tan45°- Cosec60°/Sec30° + Cos60° + Cot45°
1/2+1-2/√3/2/√3+1/2+1
=3/2-2/√3/2/√3 + 3/2
= 3√3-4/4+3√3
= 3√3-4/4 + 3√3 x 3√3-4/3√3-4
=27+16-24-√3/27-16
43-24-√3/11
[tex] \sf \implies \frac{ sin 30° + tan 45° - cosec 60° }{ sec 30° + cos 60° + cot 45°} \\ [/tex]
[tex] \sf \longrightarrow \frac{ sin 30° + tan 45° - cosec 60° }{ sec 30° + cos 60° + cot 45°} \\ [/tex]
[tex]\\ \large \sf \longrightarrow \frac{ \frac{1}{2} + 1 - \frac{2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } +\frac{1}{2} + 1} \\ [/tex]
[tex]\\\large \sf \longrightarrow \frac{ \: \frac{1 + 2 \: \: }{2} - \frac{2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } +\frac{1 + 2 \: \: }{2}} \\ [/tex]
[tex] \large\sf \longrightarrow \frac{ \: \frac{3 \: \: }{2} - \frac{2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } +\frac{3 \: \: }{2}} \\ [/tex]
[tex] \large \sf \longrightarrow \frac{ \: \frac{3 \: \sqrt{3} \: - 4 }{2 \sqrt{3} } }{ \frac{4 + 3 \sqrt{3} }{2 \sqrt{3} } }\\ [/tex]
[tex] \large\sf \longrightarrow \frac{ \: \frac{3 \: \sqrt{3} \: - 4 }{2 \sqrt{3} } }{ \frac{4 + 3 \sqrt{3} }{2 \sqrt{3} } }\\ [/tex]
[tex]\sf \longrightarrow \: \frac{3 \: \sqrt{3} \: - 4 }{2 \sqrt{3} } \times \frac{ \: \: \: \: 2 \sqrt{3} \: \: \: }{ 4 + 3 \sqrt{3}}\\ [/tex]
[tex]\sf \longrightarrow \: \frac{3 \: \sqrt{3} \: - 4 }{ \cancel{2 \sqrt{3}} } \times \frac{ \: \: \: \: \cancel{2 \sqrt{3} \: \: \:} }{ 4 + 3 \sqrt{3}}\\ [/tex]
[tex]\sf \longrightarrow \: \frac{3 \: \sqrt{3} \: - 4 } { 4 + 3 \sqrt{3}}\\ [/tex]
[tex]\sf \longrightarrow \: \frac{(3 \: \sqrt{3} \: - 4 )} { ( 3 \sqrt{3} + 4)} \times\frac{(3 \: \sqrt{3} \: - 4 )} { ( 3 \sqrt{3} - 4)} \\ [/tex]
[tex]\sf \longrightarrow \: \frac{{(3 \: \sqrt{3}) }^{2} \: + {(4)}^{2} - 2 \times 4 \times 3 \sqrt{3} } {{ ( 3 \sqrt{3}) }^{2} - {(4)}^{2} } \\ [/tex]
[tex]\sf \longrightarrow \: \frac{9 \times 3 \: + 16 - 8 \times 3 \sqrt{3} } { 9 \times 3 - 16 } \\ [/tex]
[tex]\sf \longrightarrow \: \frac{27 \: + 16 - 24 \sqrt{3} } { 27 - 16 } \\ [/tex]
[tex]\sf \longrightarrow \: \frac{43 - 24 \sqrt{3} } { 11 } \\ [/tex]
[tex]\\ \qquad{ \underline{\rule{150pt}{3pt}}} \\ [/tex]
Copyright © 2024 EHUB.TIPS team's - All rights reserved.
Answers & Comments
We know that,
Sin30° = 1/2
Tan45° = 1
Cosec60° = 2/√3
Sec30°2/√3
Cos60° = 1/2
Cot45° = 1
Given us,
Sin30° + Tan45°- Cosec60°/Sec30° + Cos60° + Cot45°
1/2+1-2/√3/2/√3+1/2+1
=3/2-2/√3/2/√3 + 3/2
= 3√3-4/4+3√3
= 3√3-4/4 + 3√3 x 3√3-4/3√3-4
=27+16-24-√3/27-16
43-24-√3/11
Verified answer
GiveN:-
[tex] \sf \implies \frac{ sin 30° + tan 45° - cosec 60° }{ sec 30° + cos 60° + cot 45°} \\ [/tex]
SolutioN:-
[tex] \sf \longrightarrow \frac{ sin 30° + tan 45° - cosec 60° }{ sec 30° + cos 60° + cot 45°} \\ [/tex]
[tex]\\ \large \sf \longrightarrow \frac{ \frac{1}{2} + 1 - \frac{2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } +\frac{1}{2} + 1} \\ [/tex]
[tex]\\\large \sf \longrightarrow \frac{ \: \frac{1 + 2 \: \: }{2} - \frac{2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } +\frac{1 + 2 \: \: }{2}} \\ [/tex]
[tex] \large\sf \longrightarrow \frac{ \: \frac{3 \: \: }{2} - \frac{2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } +\frac{3 \: \: }{2}} \\ [/tex]
[tex] \large \sf \longrightarrow \frac{ \: \frac{3 \: \sqrt{3} \: - 4 }{2 \sqrt{3} } }{ \frac{4 + 3 \sqrt{3} }{2 \sqrt{3} } }\\ [/tex]
[tex] \large\sf \longrightarrow \frac{ \: \frac{3 \: \sqrt{3} \: - 4 }{2 \sqrt{3} } }{ \frac{4 + 3 \sqrt{3} }{2 \sqrt{3} } }\\ [/tex]
[tex]\sf \longrightarrow \: \frac{3 \: \sqrt{3} \: - 4 }{2 \sqrt{3} } \times \frac{ \: \: \: \: 2 \sqrt{3} \: \: \: }{ 4 + 3 \sqrt{3}}\\ [/tex]
[tex]\sf \longrightarrow \: \frac{3 \: \sqrt{3} \: - 4 }{ \cancel{2 \sqrt{3}} } \times \frac{ \: \: \: \: \cancel{2 \sqrt{3} \: \: \:} }{ 4 + 3 \sqrt{3}}\\ [/tex]
[tex]\sf \longrightarrow \: \frac{3 \: \sqrt{3} \: - 4 } { 4 + 3 \sqrt{3}}\\ [/tex]
[tex]\sf \longrightarrow \: \frac{(3 \: \sqrt{3} \: - 4 )} { ( 3 \sqrt{3} + 4)} \times\frac{(3 \: \sqrt{3} \: - 4 )} { ( 3 \sqrt{3} - 4)} \\ [/tex]
[tex]\sf \longrightarrow \: \frac{{(3 \: \sqrt{3}) }^{2} \: + {(4)}^{2} - 2 \times 4 \times 3 \sqrt{3} } {{ ( 3 \sqrt{3}) }^{2} - {(4)}^{2} } \\ [/tex]
[tex]\sf \longrightarrow \: \frac{9 \times 3 \: + 16 - 8 \times 3 \sqrt{3} } { 9 \times 3 - 16 } \\ [/tex]
[tex]\sf \longrightarrow \: \frac{27 \: + 16 - 24 \sqrt{3} } { 27 - 16 } \\ [/tex]
[tex]\sf \longrightarrow \: \frac{43 - 24 \sqrt{3} } { 11 } \\ [/tex]
[tex]\\ \qquad{ \underline{\rule{150pt}{3pt}}} \\ [/tex]