MathLabs

Problem 5

In triangle ABC ABC , AP AP bisects ∠BAC\angle BAC with P P on BC BC , and BQ BQ bisects ∠ABC\angle ABC with Q Q on CA CA . Given ∠BAC=60∘\angle BAC=60^\circ and AB+BP=AQ+QB AB+BP=AQ+QB , determine the possible angles of ABC ABC .
Step 4 of 5: Factor the equation
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In plain words

Use sum-to-product identities.

(2cos⁡2t−1)cos⁡(3t+30∘)=0,t=x2(2\cos2t-1)\cos(3t+30^\circ)=0,\quad t=\frac x2
Detailed analysis

Rewrite with t=x/2 t=x/2 and factor by standard trigonometric identities.