Trigonometry - Study Mode
[#121] If $$frac{{sec heta - an heta }}{{sec heta + an heta }} = frac{1}{7},, heta ,$$ xa0 xa0 lies in first quadrant, then the value of $$frac{{{ ext{cosec}}, heta + {{cot }^2} heta }}{{{ ext{cosec}}, heta - {{cot }^2} heta }}$$ xa0 is:
Correct Answer
(A) $$frac{{19}}{5}$$
Explanation
Solution: $$eqalign{
& frac{{sec heta - an heta }}{{sec heta + an heta }} = frac{1}{7} cr
& 7sec heta - 7 an heta = sec heta + an heta cr
& 6sec heta = 8 an heta cr
& sin heta = frac{6}{8} = frac{3}{4} = frac{P}{H} cr
& B = sqrt {{4^2} - {3^2}} = sqrt 7 cr
& Rightarrow frac{{{ ext{cosec}}, heta + {{cot }^2} heta }}{{{ ext{cosec}}, heta - {{cot }^2} heta }} cr
& = frac{{frac{H}{P} + {{left( {frac{B}{P}}
ight)}^2}}}{{frac{H}{P} - {{left( {frac{B}{P}}
ight)}^2}}} cr
& = frac{{frac{4}{3} + {{left( {frac{{sqrt 7 }}{3}}
ight)}^2}}}{{frac{4}{3} - {{left( {frac{{sqrt 7 }}{3}}
ight)}^2}}} cr
& = frac{{frac{4}{3} + frac{7}{9}}}{{frac{4}{3} - frac{7}{9}}} cr
& = frac{{frac{{12 + 7}}{9}}}{{frac{{12 - 7}}{9}}} cr
& = frac{{19}}{5} cr} $$
[#122] The value of $$frac{{sin left( {{{78}^ circ } + heta }
ight) - cos left( {{{12}^ circ } - heta }
ight) + left( {{{ an }^2}{{70}^ circ } - { ext{cose}}{{ ext{c}}^2}{{20}^ circ }}
ight)}}{{sin {{25}^ circ }cos {{65}^ circ } + cos {{25}^ circ }sin {{65}^ circ }}}{ ext{ is:}}$$
Correct Answer
(B) -1
Explanation
Solution: $$eqalign{
& frac{{sin left( {{{78}^ circ } + heta }
ight) - cos left( {{{12}^ circ } - heta }
ight) + left( {{{ an }^2}{{70}^ circ } - { ext{cose}}{{ ext{c}}^2}{{20}^ circ }}
ight)}}{{sin {{25}^ circ }cos {{65}^ circ } + cos {{25}^ circ }sin {{65}^ circ }}} cr
& frac{{sin left( {{{78}^ circ } + heta }
ight) - sin left( {{{78}^ circ } + heta }
ight) + left( {{{ an }^2}{{70}^ circ } - {{sec }^2}{{70}^ circ }}
ight)}}{{sin {{25}^ circ }sin {{25}^ circ } + cos {{25}^ circ }sin {{25}^ circ }}} cr
& frac{{ - 1}}{{{{sin }^2}{{25}^ circ } + {{cos }^2}{{25}^ circ }}} cr
& = - 1 cr} $$
[#123] The value of $$frac{{2 an {{60}^ circ }}}{{1 + {{ an }^2}{{60}^ circ }}} = ?$$
Correct Answer
(C) sin60°
Explanation
Solution: $$eqalign{
& frac{{2 an {{60}^ circ }}}{{1 + {{ an }^2}{{60}^ circ }}} cr
& = frac{{2 imes sqrt 3 }}{{1 + 3}} cr
& = frac{{sqrt 3 }}{2} cr
& = sin {60^ circ } cr} $$
[#124] The expression (tanθ + cotθ)(secθ + tanθ)(1 - sinθ), 0° < θ < 90° is equal to:
Correct Answer
(C) cosecθ
Explanation
Solution: $$eqalign{
& left( { an heta + cot heta }
ight)left( {sec heta + an heta }
ight)(1 - sin heta ) cr
& = left( {frac{{sin heta }}{{cos heta }} + frac{{cos heta }}{{sin heta }}}
ight)left( {frac{1}{{cos heta }} + frac{{sin heta }}{{cos heta }}}
ight)left( {1 - sin heta }
ight) cr
& = left( {frac{{{{sin }^2} heta + {{cos }^2} heta }}{{cos heta sin heta }}}
ight)left( {frac{{1 + sin heta }}{{cos heta }}}
ight)left( {1 - sin heta }
ight) cr
& = left( {frac{1}{{cos heta sin heta }}}
ight)left( {frac{{1 - {{sin }^2} heta }}{{cos heta }}}
ight) cr
& = left( {frac{1}{{cos heta sin heta }}}
ight)left( {frac{{{{cos }^2} heta }}{{cos heta }}}
ight) cr
& = frac{1}{{sin heta }} cr
& = { ext{cosec}}, heta cr} $$
[#125] The value of $$frac{{4{{ an }^2}{{30}^ circ } + {{sin }^2}{{30}^ circ }{{cos }^2}{{45}^ circ } + {{sec }^2}{{48}^ circ } - {{cot }^2}{{42}^ circ }}}{{cos {{37}^ circ }sin {{53}^ circ } + sin {{37}^ circ }cos {{53}^ circ } + an {{18}^ circ } an {{72}^ circ }}},{ ext{is:}}$$
Correct Answer
(D) $$frac{{59}}{{48}}$$
Explanation
Solution: $$eqalign{
& frac{{4{{ an }^2}{{30}^ circ } + {{sin }^2}{{30}^ circ }{{cos }^2}{{45}^ circ } + {{sec }^2}{{48}^ circ } - {{cot }^2}{{42}^ circ }}}{{cos {{37}^ circ }sin {{53}^ circ } + sin {{37}^ circ }cos {{53}^ circ } + an {{18}^ circ } an {{72}^ circ }}} cr
& = frac{{4{{left( {frac{1}{{sqrt 3 }}}
ight)}^2} + {{left( {frac{1}{2}}
ight)}^2}{{left( {frac{1}{{sqrt 2 }}}
ight)}^2} + { ext{cose}}{{ ext{c}}^2}{{42}^ circ } - {{cot }^2}{{42}^ circ }}}{{cos {{37}^ circ }cos {{37}^ circ } + sin {{37}^ circ }sin {{37}^ circ } + 1}} cr
& = frac{{4 imes frac{1}{3} + frac{1}{4} imes frac{1}{2} + 1}}{{{{cos }^2}{{37}^ circ } + {{sin }^2}{{37}^ circ } + 1}} cr
& = frac{{frac{4}{3} + frac{1}{8} + 1}}{{1 + 1}} cr
& = frac{{32 + 3 + 24}}{{24 imes 2}} cr
& = frac{{59}}{{48}} cr} $$