Trigonometry MCQ Questions and Solutions with Explanations

41.
The expression $$\frac{{{{\tan }^6}\theta - {{\sec }^6}\theta + 3{{\sec }^2}\theta \,{{\tan }^2}\theta }}{{{{\tan }^2}\theta + {{\cot }^2}\theta + 2}}$$ is equal to:

A. -cos²θsin²θ

B. sec²θcosec²θ

C. -sec²θcosec²θ

D. cos²θsin²θ

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Answer & Solution

Answer: Option A

Solution:

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

42.
A coconut tree swings with the wind in such a manner that the angle covered by its trunk is 18 degrees. If the topmost portion of the tree covers a distance of 44 metres, find the length of the tree.

A. 120 metres

B. 210 metres

C. 140 metres

D. 70 metres

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43.
Simplify the following expression cosec⁴A(1 - cos⁴ A) - 2cot²A - 1

A. sin²A

B. cosec²A

C. 1

D. 0

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44.
Which of the following is equal to secA - cosA?

A. tanA.sinA

B. cosA.sinA

C. cotA.cosA

D. sinA.cotA

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45.
(sec∅ - tan∅)²(1 + sin∅)² ÷ cos²∅ = ?

A. cos²∅

B. 1

C. cot²∅

D. -1

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46.
The value cosec(67° + θ) - sec(23° - θ) + cos15°cos35°cosec55°cos60°cosec75° is:

A. $$\frac{1}{2}$$

B. 1

C. 0

D. 2

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47.
Find $$\cos \left( { - \frac{{7\pi }}{2}} \right) = ?$$

A. $$\frac{1}{2}$$

B. 1

C. -1

D. 0

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48.
If tanA - tanB - tanC = tanA.tanB.tanC, what is the value of A in terms of B and C?

A. A = B + C

B. A = 2B - 2C

C. A = B - C

D. $${\text{A}} = \frac{{{\text{B}} - {\text{C}}}}{2}$$

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49.
$$\frac{{{{\left( {1 + \sec \theta \,{\text{cosec}}\,\theta } \right)}^2}{{\left( {\sec \theta - \tan \theta } \right)}^2}\left( {1 + \sin \theta } \right)}}{{{{\left( {\sin \theta + \sec \theta } \right)}^2} + {{\left( {\cos \theta + {\text{cosec}}\,\theta } \right)}^2}}},$$ 0° < θ < 90°, is equal to:

A. 1 + sinθ

B. sinθ

C. cosθ

D. 1 - cosθ

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Answer & Solution

Answer: Option A

Solution:

$$\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} $$

50.
The value of (1 + cotθ - cosecθ)(1 + cosθ + sinθ)secθ =?

A. sinθcosθ

B. 2

C. -2

D. secθcosecθ

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Trigonometry MCQ Questions and Solutions with Explanations | Aptitude

41. The expression $$\frac{{{{\tan }^6}\theta - {{\sec }^6}\theta + 3{{\sec }^2}\theta \,{{\tan }^2}\theta }}{{{{\tan }^2}\theta + {{\cot }^2}\theta + 2}}$$ is equal to:

Answer & Solution

42. A coconut tree swings with the wind in such a manner that the angle covered by its trunk is 18 degrees. If the topmost portion of the tree covers a distance of 44 metres, find the length of the tree.

Answer & Solution

43. Simplify the following expression cosec4A(1 - cos4 A) - 2cot2A - 1

Answer & Solution

44. Which of the following is equal to secA - cosA?

Answer & Solution

45. (sec∅ - tan∅)2(1 + sin∅)2 ÷ cos2∅ = ?

Answer & Solution

46. The value cosec(67° + θ) - sec(23° - θ) + cos15°cos35°cosec55°cos60°cosec75° is:

Answer & Solution

47. Find $$\cos \left( { - \frac{{7\pi }}{2}} \right) = ?$$

Answer & Solution

48. If tanA - tanB - tanC = tanA.tanB.tanC, what is the value of A in terms of B and C?

Answer & Solution

Answer & Solution

50. The value of (1 + cotθ - cosecθ)(1 + cosθ + sinθ)secθ =?

Answer & Solution

Read More Section(Trigonometry)

41.
The expression $$\frac{{{{\tan }^6}\theta - {{\sec }^6}\theta + 3{{\sec }^2}\theta \,{{\tan }^2}\theta }}{{{{\tan }^2}\theta + {{\cot }^2}\theta + 2}}$$ is equal to:

42.
A coconut tree swings with the wind in such a manner that the angle covered by its trunk is 18 degrees. If the topmost portion of the tree covers a distance of 44 metres, find the length of the tree.

43.
Simplify the following expression cosec⁴A(1 - cos⁴ A) - 2cot²A - 1

44.
Which of the following is equal to secA - cosA?

45.
(sec∅ - tan∅)²(1 + sin∅)² ÷ cos²∅ = ?

46.
The value cosec(67° + θ) - sec(23° - θ) + cos15°cos35°cosec55°cos60°cosec75° is:

47.
Find $$\cos \left( { - \frac{{7\pi }}{2}} \right) = ?$$

48.
If tanA - tanB - tanC = tanA.tanB.tanC, what is the value of A in terms of B and C?

50.
The value of (1 + cotθ - cosecθ)(1 + cosθ + sinθ)secθ =?