KC Sinha Mathematics Solution Class 10 Chapter 8 त्रिकोणमितीय अनुपात एवम सर्वसमिकाए ( Trigonometry Ratios and Identities ) Exercise 8.4 (Q11-Q20)

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

Question 19

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

Sol :

$\left(\frac{1+\cos \theta}{\sin \theta}\right)^{2}$ 

$\frac{(1+\cos \theta)^{2}}{\sin ^{2} \theta}$ 

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

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

$\frac{1+\cos \theta}{1-\cos \theta}$ R.H.S.

Question 20

$\frac{\cos \theta}{1+\sin \theta}=\frac{1-\sin \theta}{\cos \theta}$

Sol :

$\frac{\cos \theta}{1+\sin \theta}$

परिमेयकरण करने पर

$\frac{\cos \theta}{1+\sin \theta} \times \frac{1-\sin \theta}{1-\sin \theta}$

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

$\frac{\cos \theta(1-\cos \theta)^{2}}{\cos ^{2} \theta}$

$\frac{1-\sin \theta}{\cos \theta}$ RHS