Show that: 25^ 115^ =1 2 ( 140^ -1) - Vidyadip

Question

Show that: $\sin25^\circ\cos115^\circ=\frac{1}{2}(\sin140^\circ-1)$

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Answer

$\text{LHS}=\ \sin25^\circ\cos115^\circ$ $=\ \frac{2\sin25^\circ\cos115^\circ}{2}$ We Know that $2\sin\text{A}\cos\text{B}=\sin(\text{A+B})+\sin(\text{A}-\text{B})$ $=\ \frac{1}{2}[\sin(25^\circ+115^\circ)+\sin(25^\circ-115^\circ)]$ $=\ \frac{1}{2}[\sin140^\circ+\sin(-90^\circ)]$ $\sin(-\theta)=-\sin\theta$ $\text{And},\sin(90^\circ+\theta)=\cos\theta$ $\Rightarrow\ \frac{1}{2}[\sin(90^\circ+50^\circ)-\sin90^\circ]$ $=\ \frac{1}{2}[\cos50^\circ-1]$ Also, $\cos\theta=\sin(90^\circ+\theta)$ $\cos50^\circ=\sin(90^\circ+50^\circ)=\sin140^\circ$ $\frac{1}{2}[\sin140^\circ-1]=\text{RHS}$

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