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Complementary Trigonometric Ratios

Evaluate : ...

Question

Evaluate : $tan 1 tan 2 tan 3 . . . . tan 89$

A

1

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B

-1

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C

2

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D

0

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Solution

The correct option is A 1
$t a n 1$ $t a n 2$ $t a n 3.... t a n 89$
$t a n 1 t a n 89     $ $t a n 2 t a n 88      . . . . . t a n 44 t a n 46     $ $t a n 45     $
$(1)$ $(1)$ ...... $(1)$ $(1)$
$= 1$ [ $∵ t a n θ t a n (90 - θ) = 1$ ]

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Q. Without using trigonometric tables, evaluate the follow

{tan 1}^{0} {tan 2}^{0} {tan 3}^{0} . . . . . . . . {t a n 89}^{0}

sin (50^{\circ} + θ) - cos (40^{\circ} - θ) + tan 1^{\circ} tan 15^{\circ} tan 20^{\circ} tan 70^{\circ} tan 65^{\circ} tan 89^{\circ} + sec (90^{\circ} - θ) . c o s e c θ - tan (90^{\circ} - θ) cot θ

(3^{- 1} + 4^{- 1} + 5^{- 12})^{0}

\frac{\frac{1}{2} + 3 \frac{1}{4} + 1 \frac{1}{6}}{2 \frac{1}{2} + 1 \frac{1}{6} + 6 \frac{1}{3}} + \frac{1}{120}

((10)^{0} + (12)^{0}) \times (18)^{0}

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