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प्रश्न
Find the value of the following expression:
cos 60° cos 45° – sin 60° sin 45°
योग
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उत्तर
Given: cos 60° cos 45° – sin 60° sin 45° (Use the cosine sum identity: cos (A + B) = cos A cos B – sin A sin B).
Step-wise calculation:
1. Apply the identity with A = 60°, B = 45°:
cos 60° cos 45° – sin 60° sin 45°
= cos (60° + 45°)
= cos 105°
2. Substitute exact values:
`cos 60^circ = 1/2`
`cos 45^circ = sqrt(2)/2`
`sin 60^circ = sqrt(3)/2`
`sin 45^circ = sqrt(2)/2`
3. Compute the products:
`(1/2)(sqrt(2)/2) - (sqrt(3)/2)(sqrt(2)/2)`
= `sqrt(2)/4 - sqrt(6)/4`
= `((sqrt(2) - sqrt(6)))/4`
cos 60° cos 45° – sin 60° sin 45°
= `((sqrt(2) - sqrt(6)))/4`
= `-((sqrt(6) - sqrt(2)))/4`
= – 0.258819
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