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:   $\frac{cos \ 22^o}{sin \ 68^o}$

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

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

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

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

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

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

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

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

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

$= 2$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$= 2$

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

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

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

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

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

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

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

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

$= 3 + 2 = 5$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$= 0$

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

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

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

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

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

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

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

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

$= 0$

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

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

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

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

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

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

$= 10-5 = 5$

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