Trigonometry - Study Mode

[#271] If $${ ext{0}} < { ext{A}} < {90^ circ }{ ext{,}}$$ xa0 then the value of $$frac{1}{2}cot { ext{A}}$$xa0$$left[ {frac{{1 + left( {operatorname{sec A} - { ext{tan A}}}
ight)}}{{operatorname{cosecA} left( {sec { ext{A}} - { ext{tan A}}}
ight)}}}
ight]$$ xa0 xa0 = ?
Correct Answer

(C) 1

Explanation

Solution: $$eqalign{
& { ext{According to the question,}} cr
& { ext{Put A}} = {45^ circ } cr
& Rightarrow frac{1}{2} imes cot {45^ circ } cr
& left[ {frac{{1 + left( {sec {{45}^ circ } + an {{45}^ circ }}
ight)}}{{{ ext{cosec }}{{45}^ circ }left( {sec {{45}^ circ } - an {{45}^ circ }}
ight)}}}
ight] cr
& Rightarrow frac{1}{2}left[ {frac{{1 + {{left( {sqrt 2 - 1}
ight)}^2}}}{{sqrt 2 imes left( {sqrt 2 - 1}
ight)}}}
ight] cr
& Rightarrow frac{1}{2}left[ {frac{{1 + 2 + 1 - 2sqrt 2 }}{{2 - sqrt 2 }}}
ight] cr
& Rightarrow frac{1}{2}left[ {frac{{4 - 2sqrt 2 }}{{2 - sqrt 2 }}}
ight] cr
& Rightarrow frac{1}{2} imes 2left[ {frac{{2 - sqrt 2 }}{{2 - sqrt 2 }}}
ight] cr
& Rightarrow 1 cr} $$

[#272] The value of following is : $$frac{{sin heta .operatorname{cosec} heta . an heta .cot heta }}{{{{sin }^2} heta + { ext{co}}{{ ext{s}}^2} heta }}$$ xa0 xa0 ?
Correct Answer

(D) 1

Explanation

Solution: $$eqalign{
& = frac{{sin heta .operatorname{cosec} heta . an heta .cot heta }}{{{{sin }^2} heta + { ext{co}}{{ ext{s}}^2} heta }} cr
& = frac{{sin heta imes frac{1}{{sin heta }} imes an heta imes frac{1}{{ an heta }}}}{1} cr
& = 1 cr} $$

[#273] If cosθ + secθ = $$sqrt 3 ,$$xa0 then the value of (cos 3 θ + sec 3 θ) is?
Correct Answer

(C) 0

Explanation

Solution: $$eqalign{
& { ext{cos}} heta + sec heta = sqrt 3 cr
& { ext{Cubing both sides}} cr
& { ext{co}}{{ ext{s}}^3} heta + {sec ^3} heta + 3{ ext{cos}} heta sec heta left( {{ ext{cos}} heta + sec heta }
ight) = 3sqrt 3 cr
& { ext{co}}{{ ext{s}}^3} heta + {sec ^3} heta + 3sqrt 3 = 3sqrt 3 cr
& { ext{co}}{{ ext{s}}^3} heta + {sec ^3} heta = 0 cr} $$

[#274] The value of sin 2 2° + sin 2 4° + sin 2 6° + ........ + sin 2 90° is?
Correct Answer

(A) 23

Explanation

Solution: $$eqalign{
& { ext{According to the question,}} cr
& {sin ^2}{2^ circ } + {sin ^2}{4^ circ } + {sin ^2}{6^ circ } + ..... + {sin ^2}{90^ circ } cr
& { ext{Number of terms}} cr
& = frac{{l - a}}{d} + 1 cr
& = frac{{90 - 2}}{2} + 1 cr
& = 45 cr
& { ext{But }}{sin ^2}{90^ circ } = 1 cr
& { ext{So, 22 pairs}} + {sin ^2}{90^ circ } cr
& = 22 + 1 cr
& = 23 cr} $$

[#275] If $${ ext{A}} imes { ext{tan}}left( { heta + {{150}^ circ }}
ight)$$ xa0xa0 = $${ ext{B}} imes an $$ $$left( { heta - {{60}^ circ }}
ight){ ext{,}}$$ xa0 the value of $$frac{{{ ext{A}} - { ext{B}}}}{{{ ext{A}} + { ext{B}}}}$$ xa0is?
Correct Answer

(A) $$ - frac{{sin heta }}{2}$$

Explanation

Solution: $$eqalign{
& { ext{A}} imes { ext{tan}}left( { heta + {{150}^ circ }}
ight) = { ext{B}} imes an left( { heta - {{60}^ circ }}
ight) cr
& frac{{ ext{A}}}{{ ext{B}}} = frac{{ an left( { heta - {{60}^ circ }}
ight)}}{{ an left( { heta + {{150}^ circ }}
ight)}} cr
& { ext{Put }} heta = {90^ circ } cr
& frac{{ ext{A}}}{{ ext{B}}} = frac{{ an left( {{{90}^ circ } - {{60}^ circ }}
ight)}}{{ an left( {{{90}^ circ } + {{150}^ circ }}
ight)}} cr
& frac{{ ext{A}}}{{ ext{B}}} = frac{{ an {{30}^ circ }}}{{ an left( {{{180}^ circ } + {{60}^ circ }}
ight)}} cr
& frac{{ ext{A}}}{{ ext{B}}} = frac{{ an {{30}^ circ }}}{{ an {{60}^ circ }}} cr
& frac{{ ext{A}}}{{ ext{B}}} = frac{1}{3} cr
& { ext{then, }}frac{{{ ext{A}} + { ext{B}}}}{{{ ext{A}} - { ext{B}}}} = - frac{4}{2} cr
& Rightarrow frac{{{ ext{A}} + { ext{B}}}}{{{ ext{A}} - { ext{B}}}} = - 2 cr
& Rightarrow frac{{{ ext{A}} - { ext{B}}}}{{{ ext{A}} + { ext{B}}}} = - frac{1}{2} cr
& { ext{Put in option (i)}} cr
& - frac{{sin {{90}^ circ }}}{2} = - frac{1}{2} cr
& { ext{So, option (A) is correct }} cr} $$