第 12 课 NCERT 解决方案 - 数学第一部分 - 第 2 章反三角函数 - 第 2 章杂项练习 |设置 2

第 2 章反三角函数 - 第 2 章杂项练习 |设置 1

问题 11. 证明$\tan^{-1}(\frac{\sqrt{(1+ x )}-\sqrt{(1-x)}}{\sqrt{(1+ x) }+\sqrt{(1-x)}})=\frac{\pi}{4}-\frac{1}{2} \cos^{-1}x,-\frac{1}{\sqrt2}\le x \le1$

解决方案：

Put $x=\cos2\theta$ so that, $\theta= \frac{1}{2} \cos^{-1} x.$

Then, we have :

LHS = $\tan^{-1}(\frac{\sqrt{(1+ x) }-\sqrt{(1-x)}}{\sqrt{(1+ x )}+\sqrt{(1-x)}})$

= $\tan^{-1}(\frac{\sqrt{(1+ \cos2 \theta) }-\sqrt{(1-\cos2 \theta)}}{\sqrt{(1+ \cos2 \theta) }+\sqrt{(1-\cos2 \theta)}})$

= $\tan^{-1}(\frac{\sqrt{(2 \cos^{2} \theta) }-\sqrt{(2 \sin^{2}\theta)}}{\sqrt{(2 \cos^{2}\theta) }+\sqrt{(2 \sin^{2} \theta)}})$

= $\tan^{-1}(\frac{\sqrt{2 }\cos \theta -{\sqrt{2 }\cos\theta}}{{\sqrt{2 }\cos\theta }+{\sqrt{2 }\cos \theta}})$

= $\tan^{-1}(\frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta})=\tan^{-1}(\frac{1 - \tan \theta}{1 + \tan \theta})$

$\tan^{-1} 1- \tan^{-1}(\tan \theta)$      – $[\tan^{-1}(\frac{x-y}{1+xy})]=\tan^{-1}x+\tan^{-1}y$

$=\frac{\pi}{4} -\theta = \frac{\pi}{4}-\frac{1}{2} \cos^{-1} x$  

L.H.S = R.H.S

Hence Proved

问题 12. 证明$\frac{9\pi}{8}- \frac{9}{4} \sin^{-1} \frac{1}{3}=\frac{9}{4} \sin^{-1} \frac{2\sqrt2}{3}$

解决方案：

L.H.S. = $\frac{9\pi}{8}- \frac{9}{4} \sin^{-1} \frac{1}{3}$

= $\frac {9}{4}(\frac{\pi}{2}- \sin^{-1} \frac{1}{3})$

Using $\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}$

= $\frac {9}{4}(\cos^{-1} \frac{1}{3})$      -(1)

Now, let $\cos^{-1}\frac{1}{3}=x$ Then, $\cos x =\frac{1}{3} \implies \sin x=\sqrt{1-(\frac{1}{3})^{2}}=\frac{2\sqrt{2}}{3}$

$\therefore x=\sin^{-1} \frac{2\sqrt2}{3}$ 

Using equation(1), we get,

= $\frac{9}{4}\sin^{-1} \frac{2\sqrt2}{3}$

L.H.S = R.H.S

Hence Proved

问题 13. 解决$2\tan^{-1}(\cos x)=\tan^{-1}(2\cosec x)$

解决方案：

$2\tan^{-1}(\cos x)=\tan^{-1}(2\cosec x)$

=$\tan^{-1}(\frac{2 \cos x}{1- \cos^{2}x})=\tan^{-1} (2\cosec x)$                      –$[2 \tan^{-1} x=\tan^{-1} \frac{2x}{1-x^{2}}]$

= $\frac{2\cos x} {1-cos^{2}x}=2\cosec x$

= $\frac{ 2 \cos x}{\sin^{2}}= \frac{2}{\sin x}$

= cos x/sin x

= cot x =1

Therefore, x = π/4​

问题 14. 解决$\tan^{-1} \frac{1-x}{1+x}=\frac{1}{2} \tan^{-1} x,(x>0)$

解决方案：

$\tan^{-1} \frac{1-x}{1+x}=\frac{1}{2} \tan^{-1} x$

Let x = tanθ

$\tan^{-1}(\frac{1-tan\theta}{1+tan\theta})=\frac{1}{2} \tan^{-1} tan\theta$        

$\tan^{-1}(\frac{tan\frac{\pi}{4}-tan\theta}{tan\frac{\pi}{4}+tan\theta})=\frac{1}{2} \theta$

$\tan^{-1}tan(\frac{\pi}{4}-\theta)=\frac{\theta }{2}$

π/4 – θ = θ/2

θ = π/6

So, x = tan(π/6) = 1/√3

问题 15. 解决$\sin(\tan^{-1}x),|x|<1$等于

（一种） $\frac{x}{\sqrt{(1-x^{2})}}$ (乙) $\frac{1}{\sqrt{(1-x^{2})}}$ （C） $\frac{1}{\sqrt{(1+x^{2})}}$ (四) $\frac{x}{\sqrt{(1+x^{2})}}$

解决方案：

Let tan y = x, $\sin y = \frac{x}{\sqrt{(1+x^{2})}}$

Let $\tan^{-1} x=y$ Then,

$\therefore y=\sin^{-1}(\frac{x}{\sqrt{(1+x^{2})}}) \implies \tan^{-1}x=\sin^{-1}\frac{x}{\sqrt{(1+x^{2})}}$

$\therefore \sin(\tan^{-1}x)=sin(sin^{-1}\frac{x}{\sqrt{(1+x^{2})}})=\frac{x}{\sqrt{(1+x^{2})}}$

So, the correct answer is D.

问题 16. 解决$\sin^{-1}(1- x)-2 \sin^{-1}x=\frac{\pi}{2}$ , 那么 x 等于

(A) 0, 1/2 (B) 1, 1/2 (C) 0 (D) 1/2

解决方案：

$\sin^{-1}(1- x)-2 \sin^{-1}x=\frac{\pi}{2}$

$\implies -2 \sin^{-1}x=\frac{\pi}{2} - \sin^{-1}(1-x)$

$\implies -2 \sin^{-1}x=\cos^{-1}(1-x)$        -(1)

Let $\sin^{-1} x =\theta \to \sin \theta=x$

$\cos \theta= \sqrt{1-x^{2}}$

$\therefore \theta= \cos^{-1}(\sqrt{1-x^{2}})$

$\therefore \sin^{-1} x=cos^{-1}(\sqrt{1-x^{2}})$

Therefore, from equation(1), we have

$-2cos^{-1}(\sqrt{1-x^{2}})=\cos^{-1}(1-x)$

Put x = siny then, we have:

$-2\cos^{-1}(\sqrt{1-\sin^{2} y})=\cos^{-1}(1-\sin y)$

$-2 \cos^{-1}(\cos y)=\cos^{-1}(1-\sin y)$

$-2 y=\cos^{-1}(1-\sin y)$

$1- \sin y=\cos (-2y)=\cos 2y$

$1- \sin y= 1- 2 \sin^{2} y$

$2\sin^{2} y- \sin y=0$

$\sin y(2 \sin y-1)=0$

sin y = 0 or 1/2

x = 0 or x = 1/2

But, when x = 1/2 it can be observed that:

L.H.S. = $\sin^{-1}(1-\frac{1}{2}) -2\sin^{-1} \frac{1}{2}$

= $\sin^{-1} (\frac{1}{2})-2\sin^{-1} \frac{1}{2}$

= $-\sin^{-1} \frac{1}{2}$

= $- \frac{\pi}{6} \ne\frac{\pi}{2}\ne \space R.H.S.$

x = 1/2 is not the solution of given equation.

Thus, x = 0 

Hence, the correct answer is C

问题 17. 解决$\tan ^{-1}(\frac{x}{y})-\tan ^{-1}(\frac{x-y}{x+y})$等于

(A) π /2 (B) π /3 (C) π /4 (D) -3 π /4

解决方案

$\tan^{-1} (\frac{x}{y})-\tan^{-1} \frac{x-y}{x+y}$

$\tan^{-1}[\frac{\frac{x}{y}-\frac{x-y}{x+y}}{1+\frac{x}{y} \times\frac{x-y}{x+y}}]$                –$[\tan^{-1}x+\tan^{-1}y=[\tan^{-1}(\frac{x-y}{1+xy})]]$

$\tan^{-1}[\frac {\frac {x(x+y)-y(x-y)} {y(x+y)} } {\frac {y(x+y)+x(x-y)} {y(x+y)}}]$

${\tan}^{-1}[{\frac {x^2+xy-xy+y^2} {xy+y^2+x^2-xy}}]$

${\tan}^{-1}[\frac {x^2+y^2} {x^2+y^2}]=tan^{-1}1=\frac {\pi} {4}$

Hence, the correct answer is C