Trigonometry MCQ Questions and Solutions with Explanations | Aptitude

$$\eqalign{ & \frac{{{{\tan }^6}\theta - {{\sec }^6}\theta + 3{{\sec }^2}\theta \,{{\tan }^2}\theta }}{{{{\tan }^2}\theta + {{\cot }^2}\theta + 2}} \cr & = \frac{{{{\left( {{{\tan }^2}\theta } \right)}^3} - {{\left( {{{\sec }^2}\theta } \right)}^3} + 3{{\sec }^2}\theta \,{{\tan }^2}\theta }}{{{{\tan }^2}\theta + {{\cot }^2}\theta + 2}} \cr & \left[ {{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)} \right] \cr & = \frac{{\left( {{{\tan }^2}\theta - {{\sec }^2}\theta } \right)\left( {{{\tan }^4}\theta + {{\sec }^4}\theta + {{\tan }^2}\theta {{\sec }^2}\theta + 3{{\sec }^2}\theta {{\tan }^2}\theta } \right)}}{{{{\tan }^2}\theta + {{\cot }^2}\theta + 2}} \cr & = \frac{{ - 1\left( {{{\tan }^4}\theta + {{\sec }^4}\theta + {{\tan }^2}\theta {{\sec }^2}\theta - 3{{\sec }^2}\theta {{\tan }^2}\theta } \right)}}{{{{\tan }^2}\theta + {{\cot }^2}\theta + 2}} \cr & = \frac{{ - 1\left( {{{\tan }^4}\theta + {{\sec }^4}\theta - 2{{\sec }^2}\theta {{\tan }^2}\theta } \right)}}{{{{\tan }^2}\theta + {{\cot }^2}\theta + 2}} \cr & = \frac{{ - 1\left( {{{\sec }^2}\theta - {{\tan }^2}\theta } \right)}}{{{{\tan }^2}\theta + {{\cot }^2}\theta + 2}} \cr & = \frac{{ - 1}}{{{{\left( {\tan \theta + \cot \theta } \right)}^2}}} \cr & = \frac{{ - 1}}{{{{\left( {\frac{{\sin \theta }}{{\cos \theta }} + \frac{{\cos \theta }}{{\sin \theta }}} \right)}^2}}} \cr & = \frac{{ - 1}}{{{{\left( {\frac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{\cos \theta \sin \theta }}} \right)}^2}}} \cr & = - {\left( {\cos \theta \sin \theta } \right)^2} \cr & = - {\cos ^2}\theta {\sin ^2}\theta \cr} $$

$$\eqalign{ & {\text{Arc length}} = \frac{\theta }{{180}} \times \pi r \cr & \Rightarrow 44 = \frac{{18}}{{180}} \times \frac{{22}}{7} \times r \cr & \Rightarrow r = 140\,{\text{m}} \cr} $$

$$\eqalign{ & {\text{cose}}{{\text{c}}^4}A\left( {1 - {{\cos }^4}A} \right) - 2{\cot ^2}A - 1 \cr & {\text{Here we take }}A = {45^ \circ } \cr & = {\text{cose}}{{\text{c}}^4}{45^ \circ }\left( {1 - {{\cos }^4}{{45}^ \circ }} \right) - 2{\cot ^2}{45^ \circ } - 1 \cr & = {\left( {\sqrt 2 } \right)^4}\left\{ {1 - {{\left( {\frac{1}{{\sqrt 2 }}} \right)}^4}} \right\} - 2\left( 1 \right) - 1 \cr & = 4\left( {1 - \frac{1}{4}} \right) - 2 - 1 \cr & = 4\left( {\frac{3}{4}} \right) - 2 - 1 \cr & = 3 - 2 - 1 \cr & = 0 \cr} $$

$$\eqalign{ & \sec A - \cos A \cr & = \frac{1}{{\cos A}} - \cos A \cr & = \frac{{1 - {{\cos }^2}A}}{{\cos A}} \cr & = \frac{{{{\sin }^2}A}}{{\cos A}} \cr & = \tan A.\sin A \cr} $$

$$\eqalign{ & {\left( {\sec \phi - \tan \phi } \right)^2}{\left( {1 + \sin \phi } \right)^2} \div {\cos ^2}\phi \cr & = {\left( {\frac{1}{{\cos \phi }} - \frac{{\sin \phi }}{{\cos \phi }}} \right)^2}{\left( {1 + \sin \phi } \right)^2} \div {\cos ^2}\phi \cr & = {\left( {\frac{{1 - \sin \phi }}{{\cos \phi }}} \right)^2}{\left( {1 + \sin \phi } \right)^2} \div {\cos ^2}\phi \cr & = \frac{{{{\left( {1 - \sin \phi } \right)}^2}{{\left( {1 + \sin \phi } \right)}^2}}}{{{{\cos }^2}\phi }} \times \frac{1}{{{{\cos }^2}\phi }} \cr & = \frac{{{{\left( {1 - {{\sin }^2}\phi } \right)}^2}}}{{{{\cos }^4}\phi }} \cr & = \frac{{{{\cos }^4}\phi }}{{{{\cos }^4}\phi }} \cr & = 1 \cr} $$

cosec(67° + θ) - sec(23° - θ) + cos15°cos35°cosec55°cos60°cosec75°
= sec[90° - (67° + θ)] - sec(23° - θ) + sin75°sin55°cosec55°cos60°cosec75°
= sec(23° - θ) - sec(23° - θ) + $$\frac{1}{2}$$
= $$\frac{1}{2}$$

$$\eqalign{ & \cos \left( { - \frac{{7\pi }}{2}} \right) \cr & = \cos 7 \times \frac{{180}}{2} \cr & = \cos 630 \cr & = \cos \left( {2 \times 360 - 90} \right) \cr & = \cos 90 \cr & = 0 \cr} $$

$$\eqalign{ & \tan A - \tan B - \tan C = \tan A.\tan B.\tan C \cr & \tan A - \tan A.\tan B.\tan C = \tan B + \tan C \cr & \tan A = \frac{{\tan B + \tan C}}{{1 - \tan B.\tan C}} \cr & \tan A = \tan \left( {B + C} \right) \cr & {\text{On comparision,}} \cr & A = B + C \cr} $$

$$\eqalign{ & \frac{{{{\left( {1 + \sec \theta \,{\text{cosec}}\,\theta } \right)}^2}{{\left( {\sec \theta - \tan \theta } \right)}^2}\left( {1 + \sin \theta } \right)}}{{{{\left( {\sin \theta + \sec \theta } \right)}^2} + {{\left( {\cos \theta + {\text{cosec}}\,\theta } \right)}^2}}} \cr & = \frac{{{{\left( {1 + \frac{1}{{\cos \theta \sin \theta }}} \right)}^2}{{\left( {\frac{1}{{\cos \theta }} - \frac{{\sin \theta }}{{\cos \theta }}} \right)}^2}\left( {1 + \sin \theta } \right)}}{{{{\left( {\sin \theta + \frac{1}{{\cos \theta }}} \right)}^2} + {{\left( {\cos \theta + \frac{1}{{\sin \theta }}} \right)}^2}}} \cr & = \frac{{{{\left( {\frac{{\cos \theta \sin \theta + 1}}{{\cos \theta \sin \theta }}} \right)}^2}{{\left( {\frac{{1 - \sin \theta }}{{\cos \theta }}} \right)}^2}\left( {1 + \sin \theta } \right)}}{{{{\left( {\frac{{\cos \theta \sin \theta + 1}}{{\cos \theta }}} \right)}^2} + {{\left( {\frac{{\cos \theta \sin \theta + 1}}{{\sin \theta }}} \right)}^2}}} \cr & = \frac{{{{\left( {\frac{{\cos \theta \sin \theta + 1}}{{\cos \theta \sin \theta }}} \right)}^2}{{\left( {\frac{{1 - \sin \theta }}{{\cos \theta }}} \right)}^2}\left( {1 + \sin \theta } \right)}}{{{{\left( {\cos \theta \sin \theta + 1} \right)}^2}\left( {\frac{1}{{{{\cos }^2}\theta }} + \frac{1}{{{{\sin }^2}\theta }}} \right)}} \cr & = \frac{{{{\left( {\frac{{\cos \theta \sin \theta + 1}}{{\cos \theta \sin \theta }}} \right)}^2}{{\left( {\frac{{1 - \sin \theta }}{{\cos \theta }}} \right)}^2}\left( {1 + \sin \theta } \right)}}{{{{\left( {\cos \theta \sin \theta + 1} \right)}^2}\left( {\frac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\sin }^2}\theta {{\cos }^2}\theta }}} \right)}} \cr & = {\left( {\frac{{1 - \sin \theta }}{{\cos \theta }}} \right)^2}\left( {1 + \sin \theta } \right) \cr & = \frac{{{{\left( {1 - \sin \theta } \right)}^2}\left( {1 + \sin \theta } \right)}}{{{{\cos }^2}\theta }} \cr & = \frac{{\left( {1 - \sin \theta } \right)\left( {1 + \sin \theta } \right)\left( {1 + \sin \theta } \right)}}{{\left( {1 - {{\sin }^2}\theta } \right)}} \cr & = \left( {1 + \sin \theta } \right) \cr} $$

$$\eqalign{ & \left( {1 + \cot \theta - {\text{cosec}}\,\theta } \right)\left( {1 + \cos \theta + \sin \theta } \right)\sec \theta \cr & \Rightarrow \left( {1 + \frac{{\cos \theta }}{{\sin \theta }} - \frac{1}{{\sin \theta }}} \right)\left( {1 + \cos \theta + \sin \theta } \right)\sec \theta \cr & \Rightarrow \frac{{\left( {\sin \theta + \cos \theta - 1} \right)\left( {\sin \theta + \cos \theta + 1} \right)}}{{\sin \theta .\cos \theta }} \cr & \Rightarrow \frac{{{{\left( {\cos \theta + \sin \theta } \right)}^2} - 1}}{{\sin \theta .\cos \theta }} \cr & \Rightarrow \frac{{1 + 2\sin \theta .\cos \theta }}{{\sin \theta .\cos \theta }} \cr & \Rightarrow 2 \cr} $$