(1+ sec20°)(1+sec40°)(1+sec80°)
Answers
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Given,
The trigonometric expression is = (1+ sec20°)(1+sec40°)(1+sec80°)
To find,
The simplification of the given trigonometric expression.
Solution,
We can simply solve this mathematical problem by using the following mathematical process.
Simplification :
= (1+ sec20°)(1+sec40°)(1+sec80°)
= (1+ 1/cos20°) (1+ 1/cos40°) (1+ 1/cos80°)
= {(1+cos20°)/cos20°} {(1+cos40°)/cos40°} {(1+cos80°)/cos80°}
= (2cos²10°/cos20°) (2cos²20°/cos40°)(2cos²40°/cos80°)
= {8 × (cos²10° × cos²20° × cos²40°)} / (cos20° × cos40° × cos 80°)
= 8 × cos10° × cos20° × cos40° × (cos10°/cos80°)
= 8 × cos10° × cos20° × cos40° × (cos10°/sin10°)
= 8 × cos10° × cos20° × cos40° × cot10°
= 8 × ½ × 2 × cos10° × cos20° × cos40° × cot10°
= 8 × 1/2sin10° × 2 sin10° × cos10° × cos20° × cos40° × cot10°
= 8 × 1/2sin10° × sin20° × cos20° × cos40° × cot10°
= 8 × 1/4sin10° × 2sin20° × cos20° × cos40° × cot10°
= 8 × 1/4 sin10° × sin40° × cos40° × cot10°
= 8 × 1/8 sin10° × 2sin40° × cos40° × cot10°
= 8 × 1/8sin10° × sin80° × cot10°
= 1/sin10° × cos10° × cot10°
= cot10° × cot10°
= cot²10°
Hence, the final result of the given simplification is cot²10°