Trigonometry MCQ Questions and Solutions with Explanations | Aptitude

$$\eqalign{ & \sin A = \frac{4}{5},\,\cos A = \frac{3}{5} \cr & \sin B = \frac{{15}}{{17}},\,\cos B = \frac{8}{{17}} \cr & \sin \left( {A - B} \right) \cr & = \sin A.\cos B - \cos A.\sin B \cr & = \frac{4}{5} \times \frac{8}{{17}} - \frac{3}{5} \times \frac{{15}}{{17}} \cr & = \frac{{32}}{{85}} - \frac{{45}}{{85}} \cr & = - \frac{{13}}{{85}} \cr} $$

$$\eqalign{ & {\cos ^2}\theta = 3\left( {{{\cot }^2}\theta - {{\cos }^2}\theta } \right) \cr & {\cos ^2}\theta = 3{\cos ^2}\theta \left( {\frac{1}{{{{\sin }^2}\theta }} - 1} \right) \cr & 1 = 3\left( {{\text{cose}}{{\text{c}}^2}\theta - 1} \right) \cr & \frac{1}{3} + 1 = {\text{cose}}{{\text{c}}^2}\theta \cr & \frac{2}{{\sqrt 3 }} = {\text{cosec}}\,\theta \cr & {\text{cosec }}{60^ \circ } = {\text{cosec}}\,\theta \cr & \theta = {60^ \circ } \cr & \Rightarrow {\left( {\frac{1}{2}\sec \theta + \sin \theta } \right)^{ - 1}} \cr & = {\left( {\frac{1}{2}\sec {{60}^ \circ } + \sin {{60}^ \circ }} \right)^{ - 1}} \cr & = {\left( {\frac{1}{2} \times 2 + \frac{{\sqrt 3 }}{2}} \right)^{ - 1}} \cr & = {\left( {\frac{{2 + \sqrt 3 }}{2}} \right)^{ - 1}} \cr & = \frac{2}{{2 + \sqrt 3 }} \cr & = 2\left( {2 - \sqrt 3 } \right) \cr} $$

$$\eqalign{ & \frac{{\sin A}}{{\cot A + {\text{cosec}}\,A}} - \frac{{\sin A}}{{\cot A - {\text{cosec}}\,A}} - 1 \cr & = \sin A\left[ {\frac{{\cot A - {\text{cosec}}\,A - \cot A - {\text{cosec}}\,A}}{{{{\cot }^2}A - {\text{cose}}{{\text{c}}^2}A}}} \right] - 1 \cr & = \frac{{ - 2\sin A.{\text{cosec}}\,A}}{{ - 1}} + 1 \cr & = 2 + 1 \cr & = 3 \cr} $$

sin²38° + sin²52° + sin²30° - tan²45°
= sin²38° + cos²38° + $${\left( {\frac{1}{2}} \right)^2}$$ - 1   [Here sin²θ + cos²θ = 1]
= 1 + $$\frac{1}{4}$$ - 1
= $$\frac{1}{4}$$

$$\eqalign{ & \sec \theta + \tan \theta = p \cr & \frac{{\sec \theta + \tan \theta }}{{\sec \theta - \tan \theta }} = \frac{p}{{\frac{1}{p}}} \cr & \frac{{\sec \theta + \tan \theta }}{{\sec \theta - \tan \theta }} = \frac{{{p^2}}}{1} \cr & {\text{Apply componendo and dividendo}} \cr & \frac{{\sec \theta }}{{\tan \theta }} = \frac{{{p^2} + 1}}{{{p^2} - 1}} \cr & \frac{{\sec \theta .\cos \theta }}{{\tan \theta }} = \frac{{{p^2} + 1}}{{{p^2} - 1}} \cr & \frac{{{\text{cosec }}\theta }}{1} = \frac{{{p^2} + 1}}{{{p^2} - 1}} \cr & {\text{Apply again componendo and dividendo}} \cr & \frac{{{\text{cosec }}\theta + 1}}{{{\text{cosec }}\theta - 1}} = \frac{{{p^2}}}{1} \cr} $$

$$\eqalign{ & 1 + {\cot ^2}\theta = \frac{{625}}{{49}} \cr & {\text{cose}}{{\text{c}}^2}\theta = \frac{{625}}{{49}} \cr & {\text{cosec}}\,\theta = \frac{{25}}{7} \cr & \sin \theta = \frac{7}{{25}} \cr & \cos \theta = \frac{{24}}{{25}} \cr & {\left( {\sin \theta + \cos \theta } \right)^{\frac{1}{2}}} \cr & = {\left( {\frac{7}{{25}} + \frac{{24}}{{25}}} \right)^{\frac{1}{2}}} \cr & = {\left( {\frac{{31}}{{25}}} \right)^{\frac{1}{2}}} \cr & = \frac{{\sqrt {31} }}{5} \cr} $$

cosec(65° + θ) - sec(25° - θ) + tan²20° - cosec²70°
= sec(25° - θ) - sec(25° - θ) + tan²20° - sec²20°
= tan²20° - sec²20°
= -1

$$\eqalign{ & \frac{{2\left( {{{\sin }^6}\theta + {{\cos }^6}\theta } \right) - 3\left( {{{\sin }^4}\theta + {{\cos }^4}\theta } \right)}}{{{{\cos }^4}\theta - {{\sin }^4}\theta - 2{{\cos }^2}\theta }} \cr & = \frac{{2\left( {1 - 3{{\sin }^2}\theta .{{\cos }^2}\theta } \right) - 3\left( {1 - 2{{\sin }^2}\theta .{{\cos }^2}\theta } \right)}}{{\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)\left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right) - 2{{\cos }^2}\theta }} \cr & = \frac{{2 - 6{{\sin }^2}\theta .{{\cos }^2}\theta - 3 + 6{{\sin }^2}\theta .{{\cos }^2}\theta }}{{ - \left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)}} \cr & = \frac{{ - 1}}{{ - 1}} \cr & = 1 \cr} $$

$$\eqalign{ & \frac{{{{\sec }^2}\theta }}{{{\text{cose}}{{\text{c}}^2}\theta }} + \frac{{{\text{cose}}{{\text{c}}^2}\theta }}{{{{\sec }^2}\theta }} - \left( {{{\sec }^2}\theta + {\text{cose}}{{\text{c}}^2}\theta } \right) \cr & = \frac{{{{\sec }^4}\theta + {{\cos }^4}\theta }}{{{{\sin }^2}\theta .{{\cos }^2}\theta }} - \left( {\frac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\sin }^2}\theta .{{\cos }^2}\theta }}} \right) \cr & = \frac{{\left( {{{\sin }^4}\theta - {{\sin }^2}\theta } \right) + \left( {{{\cos }^4}\theta - {{\cos }^2}\theta } \right)}}{{{{\sin }^2}\theta .{{\cos }^2}\theta }} \cr & = \frac{{ - {{\sin }^2}\theta .{{\cos }^2}\theta - {{\sin }^2}\theta .{{\cos }^2}\theta }}{{{{\sin }^2}\theta .{{\cos }^2}\theta }} \cr & = - 2 \cr & {\bf{Alternate:}} \cr & {\text{Put }}\theta = {45^ \circ } \cr & 1 + 1 - \left( {2 + 2} \right) = - 2 \cr} $$

(2cosA + 1)(2cosA - 1) = 0
⇒ (2cosA)² - (1)² = 0
⇒ 4cos²A = 1
⇒ cos²A = $$\frac{1}{4}$$
⇒ cosA = $$\frac{1}{2}$$
⇒ cosA = cos60°
⇒ A = 60°