Note: Trigonometrical Ratios of complementary angles:

For \theta < 90^o

(i)  sin \ (90^o - \theta) = cos \ \theta            (ii) cos \ (90^o - \theta) = sin \ \theta

(iii) tan \ (90^o - \theta) = cot \ \theta          (iv) cot \ (90^o - \theta) = tan \ \theta

(v)  sec \ (90^o - \theta) = cosec \ \theta       (vi) cosec \ (90^o - \theta) = sec \ \theta


Evaluate:

Question 1:   \frac{cos \ 22^o}{sin \ 68^o}

Answer:

\frac{cos \ 22^o}{sin \ 68^o} = \frac{cos \ (90^o-68^o)}{sin \ 68^o} = \frac{sin \ 68^o}{sin \ 68^o} = 1

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Question 2:   \frac{tan \ 47^o}{cot \ 43^o}

Answer:

\frac{tan \ 47^o}{cot \ 43^o} = \frac{tan \ (90^o-43^o)}{cot \ 43^o} = \frac{cot \ 43^o}{cot \ 43^o} = 1

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Question 3:   \frac{sec \ 75^o}{cosec \ 15^o}

Answer:

\frac{sec \ 75^o}{cosec \ 15^o} = \frac{sec \ (90^o-15^o)}{cosec \ 5^o} = \frac{cosec \ 15^o}{cosec \ 15^o} = 1

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Question 4:   \frac{cos \ 55^o}{sin \ 35^o}+\frac{cot \ 35^o}{tan \ 55^o}

Answer:

\frac{cos \ 55^o}{sin \ 35^o}+\frac{cot \ 35^o}{tan \ 55^o}

= \frac{cos \ (90-35)}{sin \ 35^o}+\frac{cot \ (90-35)^o}{tan \ 55^o}

= \frac{sin \ 35^o}{sin \ 35^o}+\frac{tan \ 55^o}{tan \ 55^o}

= 2

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Question 5:   cos^2 \ 40^o + cos^2 \ 50^o

Answer:

cos^2 \ 40^o + cos^2 \ 50^o

= (cos \ (90^o-50^o))^2 + cos^2 \ 50^o = sin^2 \ 50^o + cos^2 \ 50^o = 1

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Question 6:   sec^2 \ 18^o- cot^2 \ 72^o

Answer:

sec^2 \ 18^o- cot^2 \ 72^o

= (sec \ (90^o-72^o))^2 - cot^2 \ 72^o = cosec^2 \ 72^o - cot^2 \ 72^o = 1

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Question 7:   sin \ 15^o . cos \ 75^o + cos \ 15^o.sin \ 75^o

Answer:

sin \ 15^o . cos \ 75^o + cos \ 15^o.sin \ 75^o

= sin \ (90^o - 75^o) . cos \ 75^o + cos \ (90^o - 75^o).sin \ 75^o

= cos^2 \ 75 + sin^2 \ 75 = 1

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Question 8:   sin \ 42^o . sin \ 48^o - cos \ 42^o.cos \ 48^o

Answer:

sin \ 42^o . sin \ 48^o - cos \ 42^o.cos \ 48^o

= sin \ 42^o . sin \ (90^o - 42^o) - cos \ 42^o.cos \ (90^o - 42^o)

= sin \ 42^o.cos \ 42^o - cos \ 42^o. sin \ 42^o = 0

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Question 9:   sin \ (90^o-A) cos \ A + cos \ (90^o-A) sin \ A

Answer:

sin \ (90^o-A) cos \ A + cos \ (90^o-A) sin \ A = cos^2 \ A + sin^2 \ A = 1

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Question 10:   sin^2 \ 35^o + sin^2 \ 55^o

Answer:

sin^2 \ 35^o + sin^2 \ 55^o = \{sin (90-55)\}^2 + sin^2 \ 55^o

= cos^2 \ 55^o + sin ^2 \ 55^o = 1

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Question 11:   \frac{cot \ 54^o}{tan \ 36^o}+\frac{tan \ 20^o}{cot \ 70^o} - 2

Answer:

\frac{cot \ 54^o}{tan \ 36^o}+\frac{tan \ 20^o}{cot \ 70^o}

\frac{cot \ (90^o - 36^o)}{tan \ 36^o}+\frac{tan \ (90^o - 70^o)}{cot \ 70^o}

\frac{tan \ 36^o}{tan \ 36^o}+\frac{cot \ 70^o}{cot \ 70^o} = 2

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Question 12:   \frac{2 \ tan \ 53^o}{cot \ 37^o}-\frac{cot \ 80^o}{tan \ 10^o}    [2006]

Answer:

\frac{2 \ tan \ 53^o}{cot \ 37^o}-\frac{cot \ 80^o}{tan \ 10^o}

\frac{2 \ tan \ (90^o - 37^o)}{cot \ 37^o}-\frac{cot \ (90^o - 10^o)}{tan \ 10^o}

\frac{2 \ cot \ 37^o}{cot \ 37^o}-\frac{tan \ 10^o}{tan \ 10^o} = 0

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Question 13:   cos^2 \ 25^o + cos^2 \ 65^o - tan^2 \ 45^o

Answer:

cos^2 \ 25^o + cos^2 \ 65^o - tan^2 \ 45^o

= cos^2 \ (90^o-65^o) + cos^2 \ 65^o - tan^2 \ 45^o

= sin^2 \ 65^o + cos^2 \ 65^o - tan^2 \ 45^o = 0

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Question 14:   \frac{cos^2 \ 32^o + cos^2 \ 58^o}{sin^2 \ 59^o + sin^2 \ 31^o}

Answer:

\frac{cos^2 \ 32^o + cos^2 \ 58^o}{sin^2 \ 59^o + sin^2 \ 31^o}

=  \frac{cos^2 \ (90^o - 58^o) + cos^2 \ 58^o}{sin^2 \ (90^o - 31^o) + sin^2 \ 31^o}

\frac{sin^2 \ 58^o + cos^2 \ 58^o}{cos^2 \ 31^o + sin^2 \ 31^o}  = 1

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Question 15: (\frac{sin \ 77^o}{cos \ 13^o})^2 + (\frac{cos \ 77^o}{sin \ 13^o})^2 - 2 \ cos^2 \ 45^o

Answer:

(\frac{sin \ 77^o}{cos \ 13^o})^2 + (\frac{cos \ 77^o}{sin \ 13^o})^2 - 2 \ cos^2 \ 45^o

(\frac{sin \ (90^o - 13^o)}{cos \ 13^o})^2 + (\frac{cos \ (90^o - 13^o)}{sin \ 13^o})^2 - 2 \ cos^2 \ 45^o

(\frac{cos \ 13^o}{cos \ 13^o})^2 + (\frac{sin \ 13^o}{sin \ 13^o})^2 - 2 \ cos^2 \ 45^o

 = 1 + 1 - 2 \times  (\frac{1}{\sqrt{2}})^2   = 1

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Question 16:   cos^2 \ 26^o + cos \ 64^o.sin \ 26^o + \frac{tan \ 36^o}{cot \ 54^o}     [2012]

Answer:

cos^2 \ 26^o + cos \ 64^o.sin \ 26^o + \frac{tan \ 36^o}{cot \ 54^o}

= cos^2 \ 26^o + cos \ (90^o - 26^o).sin \ 26^o + \frac{tan \ (90^o - 54^o)}{cot \ 54^o}

= cos^2 \ 26^o + sin^2 \ 26^o + \frac{cot \ 54^o}{cot \ 54^o}

= 2

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Question 17:   3 .\frac{sin \ 72^o}{cos \ 18^o} - \frac{sec \ 32^o}{cosec \ 58^o}

Answer:

3 .\frac{sin \ 72^o}{cos \ 18^o} - \frac{sec \ 32^o}{cosec \ 58^o}

3 .\frac{sin \ (90^o - 18^o)}{cos \ 18^o} - \frac{sec \ (90^o - 32^o)}{cosec \ 58^o}

 = 3 .\frac{cos \ 18^o}{cos \ 18^o} - \frac{cosec \ 58^o}{cosec \ 58^o} = 3 - 1 = 2

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Question 18:   3 \ cos \ 80^o . cosec \ 10^o + 2 sin \ 59^o.sec \ 31^o     [2013]

Answer:

3 cos \ 80^o . cosec \ 10^o + 2 sin \ 59^o.sec \ 31^o

= 3 cos \ 80^o . cosec \ (90^o - 80^o) + 2 sin \ 59^o.sec \ (90^o - 59^o)

= 3 cos \ 80^o . sec \ 80^o + 2 sin \ 59^o . cosec \ 59^o

= 3 + 2 = 5

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Question 19:   \frac{sin \ 80^o}{cos \ 10^o} + sin \ 59^o . sec \ 31^o     [2007]

Answer:

\frac{sin \ 80^o}{cos \ 10^o} + sin \ 59^o . sec \ 31^o

= \frac{sin \ (90^o - 10^o)}{cos \ 10^o} + sin \ 59^o . sec \ (90^o - 59^o)

= \frac{cos \ 10^o}{cos \ 10^o} + sin \ 59^o . cosec \ 59^o = 2

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Question 20:   tan \ (55^o-A) - cot \ (35^o+A)

Answer:

tan \ (55^o-A) - cot \ (35^o+A)

= tan \ ((90^o -(35^o+A)) - cot \ (35^o+A)

= cot \ (35^o+A) - cot \ (35^o+A) = 0

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Question 21:   cosec \ (65^o+A) - sec \ (25^o-A)

Answer:

cosec \ (65^o+A) - sec \ (25^o-A)

= cosec \ (90^o - (25^o-A)) - sec \ (25^o-A)

= sec \ (25^o-A) - sec \ (25^o-A) = 0

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Question 22:   2 .\frac{tan \ 57^o}{cot \ 33^o} - \frac{cot \ 70^o}{tan \ 20^o} - \sqrt{2} \ cos \ 45^o

Answer:

2 .\frac{tan \ 57^o}{cot \ 33^o} - \frac{cot \ 70^o}{tan \ 20^o} - \sqrt{2} \ cos \ 45^o

= 2 .\frac{tan \ (90^o - 33^o)}{cot \ 33^o} - \frac{cot \ (90^o - 20^o)}{tan \ 20^o} - \sqrt{2} \ cos \ 45^o

2 .\frac{cot \ 33^o}{cot \ 33^o} - \frac{tan \ 20^o}{tan \ 20^o} - \sqrt{2} \ cos \ 45^o

 = 2 - 1 - \sqrt{2} \times \frac{1}{\sqrt{2}}

= 0

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Question 23: \frac{cot^2 \ 41^o}{tan^2 \ 49^o} - 2 \frac{sin^2 \ 75^o}{cos^2 \ 15^o}

Answer:

\frac{cot^2 \ 41^o}{tan^2 \ 49^o} - 2 \frac{sin^2 \ 75^o}{cos^2 \ 15^o}

\frac{cot^2 \ (90^o - 49^o)}{tan^2 \ 49^o} - 2 \frac{sin^2 \ (90^o - 15^o)}{cos^2 \ 15^o}

\frac{tan^2 \ 49^o}{tan^2 \ 49^o} - 2 \frac{cos^2 \ 15^o}{cos^2 \ 15^o} =-1

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Question 24: \frac{cos \ 70^o}{sin \ 20^o} + \frac{cos \ 59^o}{sin \ 31^o} - 8 \  sin^2 \ 30^o

Answer:

\frac{cos \ 70^o}{sin \ 20^o} + \frac{cos \ 59^o}{sin \ 31^o} - 8 \  sin^2 \ 30^o

= \frac{cos \ (90^o - 20^o)}{sin \ 20^o} + \frac{cos \ (90^o - 31^o)}{sin \ 31^o} - 8 \  sin^2 \ 30^o

\frac{sin \ 20^o}{sin \ 20^o} + \frac{sin \ 31^o}{sin \ 31^o} - 8  \times \frac{1}{4}

 = 0

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Question 25: 14 \ sin \ 30^o + 6 cos \ 60^o - 5 \ tan \ 45^o     [2004]

Answer:

14 \ sin \ 30^o + 6 cos \ 60^o - 5 \ tan \ 45^o

= 14 \ sin \ (90^o - 60^o) + 6 cos \ 60^o - 5 \ tan \ 45^o

= 14 \ cos \ 60^o + 6 cos \ 60^o - 5 \ tan \ 45^o

= 20 cos \ 60^o - 5 \ tan \ 45^o

= 20 \times \frac{1}{2} - 5 \times 1

= 10-5 = 5

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