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Question Number 152094 by mnjuly1970 last updated on 25/Aug/21

Answered by Olaf_Thorendsen last updated on 25/Aug/21

$$7\cos x+4\sin x+5=0\left(1\right)$$$$\mathrm{Let}t=\tan \frac{x}{2}$$$$\left(1\right):7\frac{1-t^2}{1+t^2}+4\frac{2t}{1+t^2}+5=0$$$$7(1-t^2)+8t+5(1+t^2)=0$$$$t^2-4t-6=0$$$${(t-2)}^2=10$$$$t=2\pm \sqrt{10}(=\tan \frac{\alpha }{2}\mathrm{and}\tan \frac{\beta }{2})$$$$\cot \frac{\alpha }{2}+\cot \frac{\beta }{2}=\frac{1}{2-\sqrt{10}}+\frac{1}{2+\sqrt{10}}=-\frac{2}{3}$$$$$$

Commented by mnjuly1970 last updated on 25/Aug/21

$$vratefulsitolaf..$$

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