Write the number of solutions of the equation + =2 in the interva… - Vidyadip

Question

Write the number of solutions of the equation $\tan\text{x}+\sec\text{x}=2\cos\text{x}$ in the interval $[0, 2\pi].$

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Answer

$\tan\text{x}+\sec\text{x}=2\cos\text{x}$ $1+\sin\text{x}=2\cos^2\text{x}$ $2(1-\sin^2\text{x})=1+\sin\text{x}$ $2\sin^2\text{x}+\sin\text{x}=1=0$ $(2\sin\text{x}-1)(\sin\text{x}+1)=0$ $\sin\text{x}=\frac{1}{2}$ or $-1$ $\text{x}=\frac{\pi}{6},\frac{5\pi}{6},\frac{3\pi}{2}$ Thus, there are 3 solutions in the interval $[0, 2\pi]$

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