Trigonometry - Study Mode
[#96] What is the value of the expression cos2Acos2B + sin 2 (A - B) - sin 2 (A + B)?
Correct Answer
(C) cos(2A + 2B)
Explanation
Solution: cos2Acos2B + sin 2 (A - B) - sin 2 (A + B) = cos2Acos2B + [{sin(A - B) + sin(A + B)}{sin(A - B) - sin(A + B)}] = cos2Acos2B + [(sinAcosB - cosAsinB + sinAcosB + cosAsinB)(sinAcosB - cosAsinB - sinAcosB - cosAsinB)] = cos2Acos2B + [(2sinAcosB) × (-2cosAsinB)] = cos2Acos2B - (2sinAcosA) × (2sinBcosB) = cos2Acos2B - sin2Asin2B = cos(2A + 2B) Alternate: cos2Acos2B + sin 2 (A - B) - sin 2 (A + B) = cos2Acos2B + sin2Asin2B = cos(2A + 2B)
[#97] The value of $$frac{{left( {cos {9^ circ } + sin {{81}^ circ }}
ight)left( {sec {9^ circ } + { ext{cosec}},{{81}^ circ }}
ight)}}{{sin {{56}^ circ }sec {{34}^ circ } + cos {{25}^ circ }{ ext{cosec}},{{65}^ circ }}}{ ext{ is:}}$$
Correct Answer
(C) 2
Explanation
Solution: $$eqalign{
& frac{{left( {cos {9^ circ } + sin {{81}^ circ }}
ight)left( {sec {9^ circ } + { ext{cosec}},{{81}^ circ }}
ight)}}{{sin {{56}^ circ }sec {{34}^ circ } + cos {{25}^ circ }{ ext{cosec}},{{65}^ circ }}} cr
& = frac{{left( {cos {9^ circ } + cos {9^ circ }}
ight)left( {sec {9^ circ } + sec {9^ circ }}
ight)}}{{cos {{34}^ circ }sec {{34}^ circ } + cos {{25}^ circ }sec {{25}^ circ }}} cr
& = frac{{2cos {9^ circ } imes 2sec {9^ circ }}}{{1 + 1}} cr
& = frac{{4 imes cos {9^ circ }sec {9^ circ }}}{2} cr
& = 2 cr} $$
[#98] If 7sin 2 θ + 4cos 2 θ = 5 and θ lies in the first quadrant, then what is the value of $$frac{{sqrt 3 sec heta + an heta }}{{sqrt 2 cot heta - sqrt 3 cos heta }}?$$
Correct Answer
(B) 2(1 + √2)
Explanation
Solution: $$eqalign{
& 7{sin ^2} heta + 4{cos ^2} heta = 5 cr
& 3{sin ^2} heta + 4{sin ^2} heta + 4{cos ^2} heta = 5 cr
& 3{sin ^2} heta + 4 = 5 cr
& 3{sin ^2} heta = 5 - 4 cr
& 3{sin ^2} heta = 1 cr
& {sin ^2} heta = frac{1}{3} cr
& sin heta = frac{1}{{sqrt 3 }} = frac{P}{H} cr
& B = sqrt {{{left( {sqrt 3 }
ight)}^2} - {1^2}} = sqrt 2 cr
& Rightarrow frac{{sqrt 3 sec heta + an heta }}{{sqrt 2 cot heta - sqrt 3 cos heta }} cr
& = frac{{sqrt 3 imes frac{{sqrt 3 }}{{sqrt 2 }} + frac{1}{{sqrt 2 }}}}{{sqrt 2 imes frac{{sqrt 2 }}{1} - sqrt 3 imes frac{{sqrt 2 }}{{sqrt 3 }}}} cr
& = frac{{frac{4}{{sqrt 2 }}}}{{2 - sqrt 2 }} cr
& = frac{{2sqrt 2 }}{{2 - sqrt 2 }} cr
& = frac{{2sqrt 2 }}{{2 - sqrt 2 }} imes frac{{2 + sqrt 2 }}{{2 + sqrt 2 }} cr
& = frac{{2sqrt 2 left( {2 + sqrt 2 }
ight)}}{{4 - 2}} cr
& = 2left( {sqrt 2 + 1}
ight) cr} $$
[#99] The numerical value of 1 + $$frac{1}{{{ ext{co}}{{ ext{t}}^2}{{63}^ circ }}}$$ xa0- $${ ext{se}}{{ ext{c}}^2}{27^ circ }$$ xa0+ $$frac{1}{{{{sin }^2}{{63}^ circ }}}$$ xa0- $${ ext{cose}}{{ ext{c}}^2}{27^ circ }$$ xa0 is?
Correct Answer
(D) 0
Explanation
Solution: $$eqalign{
& { ext{1 + }}frac{1}{{{ ext{co}}{{ ext{t}}^2}{{63}^ circ }}} - { ext{se}}{{ ext{c}}^2}{27^ circ }{ ext{ + }}frac{1}{{{{sin }^2}{{63}^ circ }}} - { ext{cose}}{{ ext{c}}^2}{27^ circ } cr
& Rightarrow 1 + { ext{ta}}{{ ext{n}}^2}{63^ circ } - { ext{se}}{{ ext{c}}^2}{27^ circ } + { ext{cose}}{{ ext{c}}^2}{63^ circ } - { ext{cose}}{{ ext{c}}^2}{27^ circ } cr
& Rightarrow 1 + { ext{co}}{{ ext{t}}^2}{27^ circ } - {sec ^2}{27^ circ } + { ext{se}}{{ ext{c}}^2}{27^ circ } - { ext{cose}}{{ ext{c}}^2}{27^ circ } cr
& Rightarrow 1 + { ext{co}}{{ ext{t}}^2}{27^ circ } - { ext{cose}}{{ ext{c}}^2}{27^ circ } cr
& Rightarrow 1 - 1 cr
& Rightarrow 0 cr} $$
[#100] The value of $$frac{1}{{sqrt 2 }}{ ext{sin}}frac{pi }{6}$$ . $${ ext{cos}}frac{pi }{4}$$ - $$cot frac{pi }{3}$$xa0. $${ ext{sec}}frac{pi }{6}$$ + $$frac{{5 an frac{pi }{4}}}{{12sin frac{pi }{2}}}$$ xa0 is equal to?
Correct Answer
(A) 0
Explanation
Solution: $$eqalign{
& frac{1}{{sqrt 2 }}{ ext{sin}}frac{pi }{6}{ ext{.cos}}frac{pi }{4} - cot frac{pi }{3}{ ext{.sec}}frac{pi }{6}{ ext{ + }}frac{{5 an frac{pi }{4}}}{{12sin frac{pi }{2}}} cr
& Rightarrow frac{1}{{sqrt 2 }} imes frac{1}{2} imes frac{1}{{sqrt 2 }} - frac{1}{{sqrt 3 }} imes frac{2}{{sqrt 3 }} + frac{{5 imes 1}}{{12 imes 1}} cr
& Rightarrow frac{1}{4} - frac{2}{3} + frac{5}{{12}} cr
& Rightarrow frac{{3 - 8 + 5}}{{12}} cr
& Rightarrow 0 cr
& { ext{ }} cr} $$