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| In each of the following questions, consider a right-angled triangle ABC with the right angle at B. |
1. cosec (90o − A) =
• cosec A
• sin A
• cos A
• sec A
• tan A
Answer: sec Acosec (90o − A) = 1/sin (90o − A) = 1/cos A = sec A. This cofunction property is worth remembering.
2. sec (90o − A) =
• sec A
• sin A
• cosec A
• cos A
• tan A
Answer: cosec Asec (90o − A) = 1/cos (90o − A) = 1/sin A = cosec A. This cofunction property is worth remembering.
3. cot (90o − A) =
• cos A
• sin A
• sec A
• cosec A
• tan A
Answer: tan Acot (90o − A) = 1/tan (90o − A) = 1/cot A = tan A. This cofunction property is worth remembering.
4. sin (90o − A) / cos (90o − A) =
• cosec A
• cot A
• tan A
• cot (90o − A)
• 1
Answer: cot AWe know sin (90o − A) = cos A. Similarly cos (90o − A) = sin A. Substituting these values in the expression, we get sin (90o − A)/cos (90o − A) = cos A/sin A = cot A.
5. cos (90o − A) / sin (90o − A) =
• tan A
• 1
• cot A
• tan (90o − A)
• sec (90o − A)
Answer: tan ASubstituting the values cos (90o − A) = sin A and sin (90o − A) = cos A in the expression, we get cos (90o − A)/sin (90o − A) = sin A/cos A = tan A.
6. [cosec2 A − 1] s cos (90o − A) / sin (90o − A) =
• tan A
• cot A
• 1
• sin A
• cos A
Answer: cot AThe value of cosec2 A − 1 = cot2 A. Also the value of cos (90o − A) / sin (90o − A) = sin A / cos A = tan A. Substituting these values, the answer is found to be cot2 A tan A = cot A.
7. cot A [cos (90o − A) / sin (90o − A)] =
• tan A
• cot2 A
• tan2 A
• 1
• cot A
Answer: 1The value of cos (90o − A) / sin (90o − A) = sin A / cos A = tan A. Therefore the given expression reduces to cot A tan A which equals 1.
8. [1 + sin A + sin2 A + sin3 A + sin4 A] s [
• 1
• cos A
• 0
• sin A
• tan A
Answer: 0It is virtually impossible to reduce the first term of the expression to any simple form. But the manipulation of the second term gives sec (90o − A) / cosec (90o − A) − tan (90o − A) = cosec A / sec A − cot A = cot A − cot A = 0. Hence the entire expression reduces to 0.
9. The value of tan 45o − cos 45o sin 45o is
• 1
• 0
• 1 / 2
• 3 / 4
• 1 / 4
10. The value of sin2 30o + cos2 30o is
• 0
• 1 / 2
• 3 / 2
• 1 / 4
• 1
11. The value of tan 45o + cos 0 + sin 90o is
• 2
• 3
• 1
• 1 / 2
• 0
12. The value of tan 60o cos 30o − sin 60o tan 30o is
• 0
• 1
• 1 / 2
• 3 / 2
• 2
13. The value of [1 + sin 60o + sin2 30o + sin2 60o + sin4 45o] [cos 30o − sin 60o] is
• 1
• 13 / 12
• 0
• 12 / 13
• 1 / 2
Answer: 0The value of the second term [cos 30o − sin 60o] is 0. Hence the value of the entire expression is 0, irrespective of the value of the first term.
14. (sin A + cos A)2 − 2 sin A cos A =
• 1
• 2
• 0
• tan A
• sin2 A − cos2 A
Answer: 1(sin A + cos A)2 − 2 sin A cos A = sin2 A + cos2 A = 1.
15. sin2 A − sec2 A + cos2 A + tan2 A =
• 1
• 0
• cot A
• cosec A
• cosec 2A
Answer: 0Here the terms need to be grouped properly. The given expression can be written as (sin2 A + cos2 A) − (sec2 A − tan2 A) = 1 − 1 = 0.
16. 1/(1 + cot2 A) + 1/(1 + tan2 A) =
• 0
• sin2 A
• cos2 A
• 1
• sin2 A/cos2 A
Answer: 1The expression 1/(1 + cot2 A) + 1/(1 + tan2 A) = 1/cosec2 A + 1/sec2 A = sin2 A + cos2 A = 1.
17. sin4 A − cos4 A =
• 1
• 0
• sin2 A − cos2 A
• tan2 A
• cos2 A − sin2 A
Answer: sin2 A − cos2 AWe know that (x2 − a2) can be written as (x + a)(x − a). Similarly the above expression can be written as (sin2 A + cos2 A)(sin2 A − cos2 A). The value of the first term of this expression is 1. So the answer is (sin2 A − cos2 A).
18. [(sec A − tan A)(sec A + tan A)] + [(cosec A − cot A)(cosec A + cot A)]
• 1
• 0
• 2
• 1 / 2
• sin2 A − cos2 A
Answer: 2Using the formula (x + a)(x − a) = x2 − a2, we get [sec2 A − tan2A] + [cosec2 A − cot 2A] = 1 + 1 = 2.
Try the Quiz : Practice Exercise for Trigonometry Module 2 : Trigonometric Functions and Identities
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