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Question
Prove that sin 105° + cos 105° = cos 45°
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Solution
sin 105° + cos 105° = sin(90° + 15°) + cos(90° + 15°)
= cos 15° – sin 15°
= cos(45° – 30°) sin(45° – 30°)
= (cos 45° . cos30° + sin 45° sin 30°) – (sin 45° cos 30° – cos 45° sin 30°)
= `(1/sqrt(2) * sqrt(3)/2 + 1/sqrt(2) * 1/sqrt(2)) - (1/sqrt(2) * sqrt(3)/2 - 1/sqrt(2) * 1/sqrt(2))`
= `sqrt(3)/(2sqrt(2)) + 1/(2sqrt(2)) - sqrt(3)/(2sqrt(2)) + 1/(2sqrt(2))`
= `2/(2sqrt(2))`
= `1/sqrt(2)`
= cos 45°
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