This is a classic, highly famous trick question from the Class 11 Mathematics Trigonometry chapter. When students see strange angles like 24° and 125°, they panic because these values are not on the standard 30-60-90 trigonometry table.
However, you do not need a calculator to solve this. The massive equation cos(24°) + cos(55°) + cos(125°) + cos(204°) magically collapses and perfectly cancels itself out. The final answer is exactly 0.
The Final Answer: Exactly 0.
Core Concept Used: Trigonometric Reduction Formulas (ASTC Rule).
2nd Quadrant Trick: cos(180° - θ) is always equal to -cos(θ).
3rd Quadrant Trick: cos(180° + θ) is always equal to -cos(θ).
To solve this, we must shrink the large, ugly angles (> 90°) down into small, acute angles using the supplementary angle rules (180 - θ) and (180 + θ).
Rule 1: In the 2nd Quadrant, Cosine is completely negative.
Rule 2: In the 3rd Quadrant, Cosine is also completely negative.
Let us break down the two massive angles (125° and 204°) and rewrite them using the number 180:
Step 1: Simplify cos(125°)
Step 2: Simplify cos(204°)
Step 3: Put it all together in the main equation Now, replace the large angles in the original problem with our new negative values: = cos(24°) + cos(55°) + [ -cos(55°) ] + [ -cos(24°) ]
Step 4: Cancel them out
Result = 0 + 0 = 0. The proof is complete!
The value of this entire trigonometric expression is exactly 0.
Because 125 lies in the second quadrant, where cosine is negative, you can rewrite it as cos(180 - 55), which strictly equals -cos(55).
Because the massive angles (125 and 204) convert into the exact negative versions of the small angles (-cos55 and -cos24). When added together, they perfectly act like a mirror and cancel each other out to zero.
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