Exercise 8.4
Question 11
$\frac{(1-\cos \theta)(1+\cos \theta)}{(1-\sin \theta)(1+\sin \theta)}=\tan ^{2} \theta$
Sol :
LHS
$\frac{(1-\cos \theta)(1+\cos \theta)}{(1-\sin \theta)(1+\sin \theta)}$
$\frac{1^{2}-\cos ^{2} \theta}{1^{2}-\sin ^{2} \theta}=\frac{1-\cos ^{2} \theta}{1-\sin ^{2} \theta}$
$\frac{\sin ^{2} \theta}{\cos ^{2} \theta}=\tan ^{2} \theta$
Question 12
$\frac{1}{\sec \theta+\tan \theta}=\sec \theta-\tan \theta$
Sol :
LHS
$=\frac{\sec ^{2} \theta-\tan ^{2} \theta}{\sec \theta+\tan \theta}=\frac{(\sec \theta-\tan \theta)(\sec \theta+\tan \theta)}{\sec \theta+\tan \theta}$
$=\sec \theta-\tan \theta=$ R.H.S
Question 13
$\frac{\sin ^{3} \theta+\cos ^{3} \theta}{\sin \theta+\cos \theta}+\sin \theta \cos \theta=1$
Sol :
$\frac{\sin ^{3} \theta+\cos ^{3} \theta}{\sin \theta+\cos \theta}+\sin \theta \times \cos \theta$
$a^{3}+b^{3}=(\mathrm{a}+\mathrm{b})\left(a^{2}-a b+b^{2}\right)$
$\frac{(\sin \theta+\cos \theta)\left(\sin ^{2} \theta-\sin \theta \times \cos \theta+\cos ^{2}\right)}{\sin \theta+\cos \theta}+\sin \theta \times \cos \theta$
$\sin ^{2} \theta-\sin \theta \times \cos \theta+\cos ^{2} \theta+\sin \theta \times \cos \theta$
$\sin ^{2} \theta+\cos ^{2} \theta-\sin \theta \times \cos \theta+\sin \theta \times \cos \theta$
$1-\sin \theta \times \cos \theta+\sin \theta \times \cos \theta$
=1 RHS
Type II : LCM लेकर सर्वसमिकाओं को सिद्ध करने पर आधारित प्रश्न :
Question 14
निम्नलिखित सर्वसमिकाओं को सिद्ध करें :
(i) $\sin \theta \cdot \cot \theta=\cos \theta$
Sol :
LHS
$\sin \theta \times \cot \theta$
$\sin \theta \times \frac{\cos \theta}{\sin \theta}=\cos \theta$
RHS
(ii) $\sin ^{2} \theta\left(1+\cot ^{2} \theta\right)=1$
Sol :
LHS
$\sin ^{2} \theta\left(1+\cot ^{2} \theta\right)$
$\sin ^{2} \theta \times \operatorname{cosec}^{2} \theta=(\sin \theta \times \operatorname{cosec} \theta)^{2}$
$(1)^{2}=1$=RHS
(iii) $\cos ^{2} \mathrm{~A}\left(\tan ^{2} \mathrm{~A}+1\right)=1$
Sol :
LHS
$\cos ^{2} A\left(\tan ^{2} A+1\right)$
$\cos ^{2} A \times \sec ^{2} A$
$(\cos A \times \sec A)^{2}$
$(1)^{2}=1$ RHS
(iv) $\tan ^{4} \theta+\tan ^{2} \theta=\sec ^{4} \theta-\sec ^{2} \theta$
Sol :
LHS
$\tan ^{4} \theta+\tan ^{2} \theta$
$\left(\tan ^{2} \theta\right)^{2}+\tan ^{2} \theta$
$\left(\sec ^{2} \theta-1\right)^{2}+\sec ^{2} \theta-1$
$\left(\sec ^{2} \theta\right)^{2}-2 \sec ^{2} \theta \times 1+(1)^{2}+\sec ^{2} \theta-1$
$\sec ^{4} \theta- 2\sec ^{2} \theta+1+\sec ^{2} \theta-1$☺
$\sec ^{4} \theta-\sec ^{4} \theta$
(v) $\frac{\left(1+\tan ^{2} \theta\right) \sin ^{2} \theta}{\tan \theta}=\tan \theta$
Sol :
LHS
$\frac{\left(1+\tan ^{2} \theta\right) \sin ^{2}}{\tan \theta}$
$\frac{\sec ^{2} \theta \times \sin ^{2} \theta}{\tan \theta}$
$\frac{\frac{1}{\cos ^{2} \theta} \times \sin ^{2} \theta}{\frac{\sin \theta}{\cos \theta}}$
$=\frac{1}{\cos \theta} \times \sin \theta=\frac{\sin \theta}{\cos \theta}=\tan \theta$
(vi) $\frac{\sin ^{2} \theta}{\cos ^{2} \theta}+1=\frac{\tan ^{2} \theta}{\sin ^{2} \theta}$
Sol :
LHS
$\frac{\sin ^{2} \theta}{\cos ^{2} \theta}+1$
$\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos ^{2} \theta}=\frac{1}{\cos ^{2} \theta}=\operatorname{sec}^{2} \theta$
RHS
$\frac{\tan ^{2} \theta}{\sin ^{2} \theta}=\frac{\sin ^{2} \theta}{\frac{\cos ^{2} \theta}{\sin ^{2} \theta}}$
$=\frac{1}{\cos ^{2} \theta}=\sec ^{2} \theta$
LHS=RHS
(vii) $\frac{3-4 \sin ^{2} \theta}{\cos ^{2} \theta}=3-\tan ^{2} \theta$
Sol :
LHS
$\frac{3-4 \sin ^{2} \theta}{\cos ^{2} \theta}$
$\frac{3}{\cos ^{2} \theta}-\frac{4 \sin ^{2} \theta}{\cos ^{2} \theta}$
$3 \sec ^{2} \theta-4 \tan ^{2} \theta$
$3 \tan ^{2} \theta-4 \tan ^{2} \theta$
$3-\tan ^{2} \theta $ R.H.S
(viii) $\left(1+\tan ^{2} \theta\right) \cos \theta \cdot \sin \theta=\tan \theta$
Sol :
LHS
$\left(1+\tan ^{2} \theta\right) \cos \theta \cdot \sin \theta$
$\sec ^{2} \theta \times \cos \theta \times \sin \theta$
$\frac{1}{\cos ^{2} \theta} \times \cos \theta \times \sin \theta$
$\frac{\sin \theta}{\cos \theta}=\tan \theta$
R.H.S
(ix) $\sin ^{2} \theta-\cos ^{2} \phi=\sin ^{2} \phi-\cos ^{2} \theta$
Sol :
LHS
$\sin ^{2} \theta-\cos ^{2} \phi$
$\left(1-\cos ^{2} \theta\right)-\left(1-\sin ^{2} \phi\right)$
1- $\cos ^{2} \theta-1+\sin ^{2} \phi$
$-\cos ^{2} \theta+\sin ^{2} \phi$
$\sin ^{2} \phi-\cos ^{2} \theta $ RHS
(x) $\frac{1-\tan ^{2} \theta}{\cot ^{2} \theta-1}=\tan ^{2} \theta$
Sol :
LHS
$\frac{1-\tan ^{2} \theta}{\cot ^{2} \theta-1}$
$\frac{1-\frac{\sin ^{2} \theta}{\cos ^{2} \theta}}{\frac{\cos ^{2}}{\sin ^{2}}-1}$
$\frac{\frac{\cos ^{2} \theta-\sin ^{2} \theta}{\cos ^{2} \theta}}{\frac{\cos ^{2} \theta-\sin ^{2} \theta}{\sin ^{2} \theta}}$
$\frac{\sin ^{2} \theta}{\cos ^{2} \theta}=\tan ^{2} \theta$
RHS
Question 15
निम्नलिखित सर्वसमिकाओं को सिद्ध करें :
(i) $(1-\cos \theta)(1+\cos \theta)\left(1+\cot ^{2} \theta\right)=1$
Sol :
LHS
$=(1-\cos \theta)(1+\cos \theta)\left(1+\cot ^{2} \theta\right)$
$=\left(1-\cos ^{2} \theta\right)\left(1+\cot ^{2} \theta\right)$
$=\sin ^{2} \theta \times \operatorname{cosec}^{2} \theta$
=1
(ii) $\frac{(1+\sin \theta)^{2}+(1-\sin \theta)^{2}}{2 \cos ^{2} \theta}=\frac{1+\sin \theta}{1-\sin ^{2} \theta}$
Sol :
LHS
$\frac{(1+\sin \theta)^{2}+(1-\sin \theta)^{2}}{2 \cos ^{2} \theta}$
$\frac{\left(1^{2}+\sin ^{2} \theta+2 \times 1 \times \sin \theta\right)+\left(1^{2}+\sin ^{2} \theta-2 \times 1 \times \sin \theta\right)}{2 \cos ^{2} \theta}$
$\frac{1+\sin ^{2} \theta+2 \sin \theta+1+\sin ^{2} \theta-2 \sin \theta}{2 \cos ^{2} \theta}$
$\frac{2+2 \sin ^{2} \theta}{2 \cos ^{2} \theta}=\frac{2\left(1+\sin ^{2} \theta\right)}{2\left(1-\sin ^{2} \theta\right)}$
$\frac{1-\sin ^{2} \theta}{1+\sin ^{2} \theta}$
RHS
(iii) $\frac{\cos ^{2} \theta(1-\cos \theta)}{\sin ^{2} \theta(1-\sin \theta)}=\frac{1+\sin \theta}{1+\cos \theta}$
Sol :
LHS
$\frac{\cos ^{2} \theta(1-\cos \theta)}{\sin ^{2} \theta(1-\sin \theta)}$
$\frac{\left(1-\sin ^{2} \theta\right)(1-\cos \theta)}{\left(1-\cos ^{2} \theta\right)(1-\sin \theta)}$
$\frac{(1-\sin \theta)(1+\sin )(1-\cos \theta)}{(1-\operatorname{cos} \theta)(1+\cos \theta)(1-\sin \theta)}$
$\frac{1+\sin \theta}{1+\cos \theta}$ RHS
(iv) $(\sin \theta-\cos \theta)^{2}=1-2 \sin \theta \cdot \cos \theta$
Sol :
LHS
$(\sin \theta-\cos \theta)^{2}$
$\sin ^{2} \theta+\cos ^{2} \theta-2 \times \sin \theta \times \cos \theta$
$\sin ^{2} \theta+\cos ^{2} \theta-2 \sin \theta \times \cos \theta$
$1-2 \sin \theta \times \cos \theta$
R.H.S
(v) $(\sin \theta+\cos \theta)^{2}+(\sin \theta-\cos \theta)^{2}=2$
Sol :
LHS
$(\sin \theta+\cos \theta)^{2}+(\sin \theta-\cos \theta)^{2}=2$
$\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \times \cos \theta+\sin ^{2} \theta+\cos ^{2} \theta-2 \sin \theta \times \cos \theta$
$\sin ^{2} \theta+\cos ^{2} \theta+\sin ^{2} \theta+\cos ^{2} \theta$
$2 \sin ^{2} \theta+2 \cos ^{2} \theta$
$2\left(\sin ^{2} \theta+\cos ^{2} \theta\right)$
2×1=2 RHS
(vi) $(a \sin \theta+b \cos \theta)^{2}+(a \cos \theta-b \sin \theta)^{2}=a^{2}+b^{2}$
Sol :
LHS
$(a \sin \theta+b \cos \theta)^{2}+(a \cos \theta-b \sin \theta)^{2}$
$(a \sin \theta)^{2}+(b \cos \theta)^{2}+2 \times a \sin \theta \times b \cos \theta+(a \cos \theta)^{2}+(b \sin \theta)^{2}-2 a \cos \theta \times b \sin \theta$
$a^{2} \times \sin ^{2} \theta+b^{2} \times \cos ^{2} \theta+2 a \sin \theta \times b \cos \theta+a^{2} \times \cos ^{2} \theta+b^{2} \times \sin ^{2} \theta-2 a \cos \theta \times b \sin \theta$
$a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta+a^{2} \cos ^{2} \theta+b^{2} \sin ^{2} \theta$
$\left(a^{2} \sin ^{2} \theta+a^{2} \cos ^{2} \theta\right)+\left(b^{2} \cos ^{2} \theta+b^{2} \sin ^{2} \theta\right)$
$a^{2}\left(\sin ^{2} \theta+\cos ^{2} \theta\right)+b^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)$
$a^{2} \times 1+b^{2} \times 1$
$a^{2}+b^{2}$
RHS
(vii) $\cos ^{4} A+\sin ^{4} A+2 \sin ^{2} A \cdot \cos ^{2} A=1$
Sol :
LHS
$\cos ^{4} A+\sin ^{4} A+2 \sin ^{2} A \times \cos ^{2} A$
$\left(\cos ^{2} A\right)^{2}-\left(\sin ^{2} A\right)^{2}+2 \sin ^{2} A \times \cos ^{2} A$
$\left(\cos ^{2} A+\sin ^{2} A\right) 2 \sin ^{2} A \times \cos ^{2} A$
$=(1)^{2}$=1 RHS
(viii) $\sin ^{4} A-\cos ^{4} A=2 \sin ^{2} A-1=1-2 \cos ^{2} A=\sin ^{2} A-\cos ^{2} A$
Sol :
LHS
$\sin ^{4} A-\cos ^{4} A$
$\left(\sin ^{2} A\right)^{2}-\left(\cos ^{2} A\right)^{2}$
$\left(\sin ^{2} A+\cos ^{2} A\right)\left(\sin ^{2} A-\cos ^{2} A\right)$
$=\sin ^{2} A-\cos ^{2} A$
$=\sin ^{2} A-\left(1-\sin ^{2} A\right)$
$=\sin ^{2} A-1+\sin ^{2} A$
$=2 \sin ^{2} A-1$
$=2\left(-\cos ^{2} A\right)-1$
$=2-2 \cos ^{2} A-1=1-2 \cos ^{2} A$
=RHS
(ix) $\cos ^{4} \theta-\sin ^{4} \theta=\cos ^{2} \theta-\sin ^{2} \theta=2 \cos ^{2} \theta-1$
Sol :
LHS
$\cos ^{4} \theta-\sin ^{4} \theta=\cos ^{2} \theta-\sin ^{2} \theta=2 \cos ^{2} \theta-1$
$\cos ^{4} \theta-\sin ^{4} \theta=\left(\cos ^{2} \theta\right)^{2}-\left(\sin ^{2} \theta\right)^{2}$
$\left(\cos ^{2} \theta+\sin ^{2} \theta\right)\left(\cos ^{2} \theta-\sin ^{2} \theta\right)$
$\cos ^{2} \theta-\sin ^{2} \theta$
$\cos ^{2} \theta-\left(1-\cos ^{2} \theta\right)$
$\cos ^{2} \theta-1+\cos ^{2} \theta$
$2 \cos ^{2} \theta-1$
RHS
(x) $2 \cos ^{2} \theta-\cos ^{4} \theta+\sin ^{4} \theta=1$
Sol :
LHS
$2 \cos ^{2} \theta-\cos ^{4} \theta+\sin ^{4} \theta$
$2 \cos ^{2} \theta-\left(\cos ^{2} \theta\right)^{2}+\left(\sin ^{2} \theta\right)^{2}$
$2 \cos ^{2} \theta-\left(\cos ^{2} \theta+\sin ^{2} \theta\right)$
$2 \cos ^{2} \theta-\cos ^{2} \theta+\sin ^{2} \theta$
$2 \cos ^{2} \theta-\cos ^{2} \theta+1-\cos ^{2} \theta$
$2 \cos ^{2} \theta-2 \cos ^{2} \theta+1$
0+1=1
(xi) $1-2 \cos ^{2} \theta+\cos ^{4} \theta=\sin ^{4} \theta$
Sol :
LHS
$1-2 \cos ^{2} \theta+\cos ^{4} \theta$
$1^{2}-2 \times 1 \times \cos ^{2} \theta+\left(\cos ^{2}\right)^{2}$
$\left(1-\cos ^{2} \theta\right)^{2}$
$\left(\sin ^{2} \theta\right)^{4}=\sin ^{4} \theta$
RHS
(xii) $1-2 \sin ^{2} \theta+\sin ^{4} \theta=\cos ^{4} \theta$
Sol :
LHS
$1-2 \sin ^{2} \theta+\sin ^{4} \theta$
$1^{2}-2 \times 1 \times \sin ^{2} \theta+\left(\sin ^{2}\right)^{2}$
$\left(1-\sin ^{2} \theta\right)^{2}$
$\left(\cos ^{2} \theta\right)^{2}=\cos ^{4} \theta$
RHS
Question 16
निम्नलिखित सर्वसमिकाओं को सिद्ध करें :
(i) $\sec ^{2} \theta+\operatorname{cosec}^{2} \theta=\sec ^{2} \theta \cdot \operatorname{cosec}^{2} \theta$
Sol :
LHS
$\sec ^{2} \theta+\operatorname{cosec}^{2} \theta$
$\frac{1}{\cos ^{2} \theta}+\frac{1}{\sin ^{2} \theta}$
$\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos ^{2} \theta \times \sin ^{2} \theta}$
$\frac{1}{\cos ^{2} \theta \times \sin ^{2} \theta}$
$=\frac{1}{\frac{1}{\sec ^{2} \theta} \times \frac{1}{\operatorname{cosec}^{2} \theta}}$
$\sec ^{2} \theta \times \operatorname{cosec}^{2} \theta$
RHS
(ii) $\frac{\cos ^{2} \theta}{\sin \theta}+\sin \theta=\operatorname{cosec} \theta$
Sol :
LHS
$\frac{\cos ^{2} \theta}{\sin \theta}+\sin \theta$
$\frac{1-\sin ^{2} \theta}{\sin \theta}+\sin \theta$
$\frac{1}{\sin \theta}-\frac{\sin ^{2} \theta}{\sin \theta}+\sin \theta$
$\frac{1}{\sin \theta}-\sin \theta+\sin \theta$
cosec θ
RHS
(iii) $\cot \theta+\tan \theta=\operatorname{cosec} \theta \cdot \sec \theta$
Sol :
LHS
$\cot \theta+\tan \theta$
$\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}$
$\frac{\cos ^{2} \theta+\sin ^{2} \theta}{\sin \theta \times \cos \theta}$
$\frac{1}{\sin \theta \times \cos \theta}$
$\frac{1}{\frac{1}{\sin \theta} \times \frac{1}{\cos \theta}}$
$\operatorname{cosec} \theta \times \sec \theta$
RHS
Question 17
$\frac{1-\sin \theta}{1+\sin \theta}=\left(\frac{1-\sin \theta}{\cos \theta}\right)^{2}$
Sol :
LHS
$\frac{1-\sin \theta}{1+\sin \theta}$
परिमेयकरण करने पर
$\frac{1-\sin \theta}{1+\sin \theta} \times \frac{1-\sin \theta}{1-\sin \theta}$
$\frac{(1-\sin \theta)^{2}}{(1)^{2}-(\sin \theta)^{2}}$
$\frac{(1-\sin \theta)^{2}}{1-\sin ^{2} \theta}$
$\frac{(1-\sin \theta)^{2}}{\cos ^{2} \theta$
$\left(\frac{1-\sin \theta}{\cos \theta}\right)^{2}$
RHS
Question 18
$\frac{1-\cos \theta}{1+\cos \theta}=\left(\frac{1-\cos \theta}{\sin \theta}\right)^{2}$
Sol :
LHS
$\frac{1-\cos \theta}{1+\cos \theta}$
परिमेयकरण करने पर
$\frac{1-\cos \theta}{1+\cos \theta} \times \frac{1-\cos \theta}{1-\cos \theta}$
$\frac{(1-\cos \theta)^{2}}{(1)^{2}-(\cos \theta)^{2}}$
$\frac{(1-\cos \theta)^{2}}{1-\cos ^{2} \theta}$
$\frac{(1-\cos \theta)^{2}}{\sin ^{2} \theta}$
$\left(\frac{1-\cos \theta}{\sin \theta}\right)^{2}$ RHS
I liked it 😊😊😊
ReplyDelete