111.
The value of ÃÂÃÂ - ÃÂÃÂ - ÃÂÃÂ is?
Solution:
$$\eqalign{ & = {\sec ^2}{17^ \circ } - \frac{1}{{{{\tan }^2}{{73}^ \circ }}} - \sin {17^ \circ }\sec {73^ \circ } \cr & = {\sec ^2}{17^ \circ } - {\cot ^2}{73^ \circ } - \sin {17^ \circ }\sec \left( {{{90}^ \circ } - {{17}^ \circ }} \right) \cr & = {\sec ^2}{17^ \circ } - {\cot ^2}\left( {{{90}^ \circ } - {{17}^ \circ }} \right) - \sin {17^ \circ }\cos ec{17^ \circ } \cr & = {\sec ^2}{17^ \circ } - {\tan ^2}{17^ \circ } - 1 \cr & = 1 - 1\left[ {\because {{\sec }^2}\theta - {{\tan }^2}\theta = 1} \right] \cr & = 0 \cr} $$
112.
If ÃÂÃÂ ÃÂÃÂ = ÃÂÃÂ the value of ÃÂÃÂ is?
Solution:
$$\eqalign{ & {\text{A}} \times {\text{tan}}\left( {\theta + {{150}^ \circ }} \right) = {\text{B}} \times \tan \left( {\theta - {{60}^ \circ }} \right) \cr & \frac{{\text{A}}}{{\text{B}}} = \frac{{\tan \left( {\theta - {{60}^ \circ }} \right)}}{{\tan \left( {\theta + {{150}^ \circ }} \right)}} \cr & {\text{Put }}\theta = {90^ \circ } \cr & \frac{{\text{A}}}{{\text{B}}} = \frac{{\tan \left( {{{90}^ \circ } - {{60}^ \circ }} \right)}}{{\tan \left( {{{90}^ \circ } + {{150}^ \circ }} \right)}} \cr & \frac{{\text{A}}}{{\text{B}}} = \frac{{\tan {{30}^ \circ }}}{{\tan \left( {{{180}^ \circ } + {{60}^ \circ }} \right)}} \cr & \frac{{\text{A}}}{{\text{B}}} = \frac{{\tan {{30}^ \circ }}}{{\tan {{60}^ \circ }}} \cr & \frac{{\text{A}}}{{\text{B}}} = \frac{1}{3} \cr & {\text{then, }}\frac{{{\text{A}} + {\text{B}}}}{{{\text{A}} - {\text{B}}}} = - \frac{4}{2} \cr & \Rightarrow \frac{{{\text{A}} + {\text{B}}}}{{{\text{A}} - {\text{B}}}} = - 2 \cr & \Rightarrow \frac{{{\text{A}} - {\text{B}}}}{{{\text{A}} + {\text{B}}}} = - \frac{1}{2} \cr & {\text{Put in option (i)}} \cr & - \frac{{\sin {{90}^ \circ }}}{2} = - \frac{1}{2} \cr & {\text{So, option (A) is correct }} \cr} $$
113.
The value of 4sin230ÃÂÃÂÃÂð + 3cot260ÃÂÃÂÃÂð is:
Solution:
$$\eqalign{ & 4{\sin ^2}{30^ \circ } + 3{\cot ^2}{60^ \circ } \cr & = 4 \times {\left( {\frac{1}{2}} \right)^2} + 3{\left( {\frac{1}{{\sqrt 3 }}} \right)^2} \cr & = 4 \times \frac{1}{4} + 3 \times \frac{1}{3} \cr & = 1 + 1 \cr & = 2 \cr} $$
114.
The value of ÃÂÃÂ - ÃÂÃÂ - ÃÂÃÂ is?
Solution:
$$\eqalign{ & = {\sec ^2}{17^ \circ } - \frac{1}{{{{\tan }^2}{{73}^ \circ }}} - \sin {17^ \circ }\sec {73^ \circ } \cr & = {\sec ^2}{17^ \circ } - {\cot ^2}{73^ \circ } - \sin {17^ \circ }\sec \left( {{{90}^ \circ } - {{17}^ \circ }} \right) \cr & = {\sec ^2}{17^ \circ } - {\cot ^2}\left( {{{90}^ \circ } - {{17}^ \circ }} \right) - \sin {17^ \circ }\cos ec{17^ \circ } \cr & = {\sec ^2}{17^ \circ } - {\tan ^2}{17^ \circ } - 1 \cr & = 1 - 1\left[ {\because {{\sec }^2}\theta - {{\tan }^2}\theta = 1} \right] \cr & = 0 \cr} $$
115.
The value of ÃÂÃÂ ÃÂÃÂ is:
- (A) tanθ.cosθ
- (B) secθ.sinθ
- (C) secθ.tanθ
- (D) sinθ.cosθ
Solution:
$$\eqalign{ & \frac{{\sec \theta \left( {\sin \theta - 2{{\sin }^3}\theta } \right)}}{{2{{\cos }^3}\theta - \cos \theta }} \cr & \frac{{\sec \theta .\sin \theta \left( {1 - 2{{\sin }^2}\theta } \right)}}{{\cos \theta \left( {2{{\cos }^2}\theta - 1} \right)}} \cr & \frac{{\sec \theta .\sin \theta \times \cos 2\theta }}{{\cos \theta \times \cos 2\theta }} \cr & \sec \theta .\tan \theta \cr} $$
116.
If tanÃÂÃÂÃÂø - cotÃÂÃÂÃÂø = 0 find the value of sinÃÂÃÂÃÂø + cosÃÂÃÂÃÂø ?
Solution:
$$\eqalign{ & {\text{tan}}\theta - \cot \theta = 0 \cr & {\bf{Shortcut\,\, method:}} \cr & {\text{Put }}\theta = {45^ \circ } \cr & {\text{tan }}{45^ \circ } - \cot {45^ \circ } = 0 \cr & 1 - 1 = 0 \cr & 0 - 0({\text{matched}}) \cr & So,\theta = {45^ \circ } \cr & \Rightarrow \sin \theta + \cos \theta \cr & \Rightarrow \sin {45^ \circ } + \cos {45^ \circ } \cr & \Rightarrow \frac{1}{{\sqrt 2 }} + \frac{1}{{\sqrt 2 }} \cr & \Rightarrow \sqrt 2 \cr} $$
117.
The value of sin265ÃÂÃÂÃÂð + sin225ÃÂÃÂÃÂð + cos235ÃÂÃÂÃÂð + cos255ÃÂÃÂÃÂð is?
Solution:
$$\eqalign{ & \Rightarrow {\sin ^2}{65^ \circ } + {\sin ^2}{25^ \circ } + {\cos ^2}{35^ \circ } + {\cos ^2}{55^ \circ } \cr & \Rightarrow {\sin ^2}{65^ \circ } + {\sin ^2}\left( {{{90}^ \circ } - {{65}^ \circ }} \right) + \left[ {{{\cos }^2}{{35}^ \circ } + {{\cos }^2}\left( {{{90}^ \circ } - {{35}^ \circ }} \right)} \right] \cr & \Rightarrow \left( {{{\sin }^2}{{65}^ \circ } + {{\cos }^2}{{65}^ \circ }} \right) + \left( {{{\cos }^2}{{35}^ \circ } + si{n^2}{{35}^ \circ }} \right) \cr & \Rightarrow 1 + 1 \cr & \Rightarrow 2 \cr} $$
118.
If ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ the value of ÃÂÃÂ + ÃÂÃÂ is?
Solution:
$$\eqalign{ & {\text{2cos}}\theta - \sin \theta = \frac{1}{{\sqrt 2 }} \cr & {\text{When,}} \cr & ax \mp by = m \cr & {\text{then, }}bx \mp ay = \sqrt {{a^2} + {b^2} - {m^2}} \cr & 2\cos \theta - \sin \theta = \frac{1}{{\sqrt 2 }} \cr & \Rightarrow \cos \theta + 2\sin \theta = \sqrt {4 + 1 - \frac{1}{2}} \cr & \Rightarrow \cos \theta + 2\sin \theta = \frac{3}{{\sqrt 2 }} \cr} $$
119.
The value of the following is : ÃÂÃÂ + ÃÂÃÂ - ÃÂÃÂ = ?
Solution:
$$\eqalign{ & {\left( {\frac{{{\text{sin 4}}{{\text{7}}^ \circ }}}{{\cos {{43}^ \circ }}}} \right)^2} + {\left( {\frac{{\cos {{43}^ \circ }}}{{{\text{sin }}{{47}^ \circ }}}} \right)^2} - 4{\text{co}}{{\text{s}}^2}{45^ \circ } \cr & = {\left( {\frac{{{\text{cos 4}}{{\text{3}}^ \circ }}}{{\cos {{43}^ \circ }}}} \right)^2} + {\left( {\frac{{sin{{47}^ \circ }}}{{{\text{sin }}{{47}^ \circ }}}} \right)^2} - 4 \times \frac{1}{2} \cr & = 1 + 1 - 2 \cr & = 0 \cr & {\bf{Note:}} \cr & \left( {\sin \left( {{{90}^ \circ } - \theta } \right)} \right) = \cos \theta \cr} $$
120.
The value of sec228ÃÂÃÂÃÂð - cot262ÃÂÃÂÃÂð + sin260ÃÂÃÂÃÂð + cosec230ÃÂÃÂÃÂð is equal to:
Solution:
$$\eqalign{ & {\sec ^2}{28^ \circ } - {\cot ^2}{62^ \circ } + {\sin ^2}{60^ \circ } + {\text{cose}}{{\text{c}}^2}{30^ \circ } \cr & = {\sec ^2}{28^ \circ } - {\tan ^2}{28^ \circ } + {\sin ^2}{60^ \circ } + {\text{cose}}{{\text{c}}^2}{30^ \circ } \cr & = 1 + {\left( {\frac{{\sqrt 3 }}{2}} \right)^2} + {\left( 2 \right)^2} \cr & = 1 + \frac{3}{4} + 4 \cr & = \frac{{23}}{4} \cr} $$