Question

Without using trigonometric tables evaluate the following:
cot 30° cosec 30° 2 cos 0°
sec 30° tan 45° sin 300 + cos 45º​

Answer 1

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Answer 2

Evaluation of cot 30° cosec 30° 2 cos 0°  sec 30° tan 45° sin 300 + cos 45º​ is done as follows.

Given,

cot 30° cosec 30° 2 cos 0°  sec 30° tan 45° sin 300 + cos 45º​

We know the trigonometric ratios:

sin = 1/coses

cos = 1/sec

tan = 1/cot

tan = sin/cos

cot = cos/sin

Now we have,

cot 30° cosec 30° 2 cos 0°  sec 30° tan 45° sin 300 + cos 45º​

= (cos 30° / sin 30°) × (1 / sin 30°) × (2 cos 0°) × (1 / cos 30°) × (sin 45° / cos 45°) × sin (360°  - 60° ) + cos 45º

= 2 cos ​0° / sin² 30° × (sin 45° / cos 45°) × (-sin 60° ) + cos 45º

= (2 cos ​0° / sin² 30° ) × (sin 45° / cos 45°) × (-sin 60° ) + cos 45º

= (2 cos ​0° / sin² 30° ) × (sin (90° - 45°) / cos 45°) × (-sin 60° ) + cos 45º

= (2 cos ​0° / sin² 30° ) × (cos 45° / cos 45°) × (-sin 60° ) + cos 45º

= (2 cos ​0° / sin² 30° )  × (-sin 60° ) + cos 45º

= - (2 cos ​0° / sin² 30° )  × (2 sin 30° cos 30° ) + cos 45º

= - (2 cos ​0° / sin 30° )  × (2 cos 30° ) + cos 45º

= - 4 / sin 30° (cos 0°   × cos 30°) + cos 45º

= - 4 / sin 30° [ cos (30° + 0°) + cos (30° - 0°) ] /2 + cos 45º

= - 4 / sin 30° [ cos 30°  + cos 30° ] /2 + cos 45º

= - 4 / sin 30° [ 2 cos 30°  ] /2 + cos 45º

= - 4 / sin 30° × cos 30°  + cos 45º