Solve: sec x- tan x=√3 .

Question

Solve: [latex] \sec x- an x=\sqrt{3} [/latex].

Accepted Answer

We have

[latex]\begin{aligned}\sec x- an x=\sqrt{3} & \Rightarrow \frac{1}{\cos x}-\frac{\sin x}{\cos x}=\sqrt{3} \Rightarrow(1-\sin x)=\sqrt{3} \cos x\ \& \Rightarrow \sqrt{3} \cos x+\sin x=1 .\end{aligned}[/latex]

Thus, the given equation becomes

[latex]\sqrt{3} \cos x+\sin x=1.\qquad \qquad ...(i)[/latex]

Dividing both sides of (i) by [latex] \sqrt{(\sqrt{3})^{2}+1^{2}} [/latex], i.e., by 2 , we get

[latex]\begin{aligned}& \frac{\sqrt{3}}{2} \cos x+\frac{1}{2} \sin x=\frac{1}{2} \ \\Rightarrow & \cos x \cos \frac{\pi}{6}+\sin x \sin \frac{\pi}{6}=\frac{1}{2} \ \\Rightarrow & \cos \left(x-\frac{\pi}{6} ight)=\cos \frac{\pi}{3} \ \\Rightarrow & \left(x-\frac{\pi}{6} ight)=2 n \pi \pm \frac{\pi}{3}, ext { where } n \in I [\because \cos heta=\cos \alpha \Rightarrow heta=2 n \pi \pm \alpha]\ \\Rightarrow & \left(x-\frac{\pi}{6} ight)=\left(2 n \pi+\frac{\pi}{3} ight) ext { or }\left(x-\frac{\pi}{6} ight)=\left(2 n \pi-\frac{\pi}{3} ight) \ \\Rightarrow & x=2 n \pi+\left(+\frac{\pi}{3}+\frac{\pi}{6} ight) ext { or } x=2 n \pi+\left(-\frac{\pi}{3}+\frac{\pi}{6} ight), ext { where } n \in I \ \\Rightarrow & x=\left(2 n \pi+\frac{\pi}{2} ight) ext { or } x=\left(2 n \pi-\frac{\pi}{6} ight), ext { where } n \in I \ \\Rightarrow & x=\left(2 n \pi-\frac{\pi}{6} ight), ext { where } n \in I \ \ & \left[ \because \sec x ext{ is not defined when } x =(2n\pi + \frac{\pi}{2}) ight].\end{aligned}[/latex]

Hence, the general solution is [latex] x=\left(2 n \pi-\frac{\pi}{6} ight) [/latex], where [latex] n \in I [/latex].

Question 17.2.13: Solve: sec x- tan x=√3 ....

Related Answered Questions

Find the principal solutions of each of the following equations: (i) sin x=1/2 (ii) cos x=1/√2 (iii) tan x=1/√3 ...

Verified Answer:

Solve: √3cos x-sin x=1 . ...

Verified Answer:

Find the general solution for each of the following equations: (i) cos 4x=cos 2x (ii) cos 3x=sin 2x (iii) sin 3x+cos 2x=0 (iv) sin mx+sin nx=0 ...

Verified Answer:

Solve: 2cos² x+3sin x=0 . ...

Verified Answer:

Find the general solution of the equation sin 2x+sin 4x+sin 6x=0 . ...

Verified Answer:

Find the general solution of each of the equations: (i) 4sin² x=1 (ii) 2cos² x=1 (iii) cot² x=3 ...

Verified Answer:

Find the general solution of each of the equations: (i) sin 2x=-1/2 (ii) tan 3x=-1 ...

Verified Answer:

Find the general solution of each equation: (i) √3 cot x+1=0 (ii) cosec x+√2=0 ...

Verified Answer:

In each of the following, find the general value of x satisfying the equation: (i) sin x=-√3/2(ii) cos x=-1/2(iii) cot x=-√3 ...

Verified Answer:

Find the general value of x for which √3 cosec x=2 . ...

Verified Answer: