KC Sinha Mathematics Solution Class 10 Chapter 8 त्रिकोणमितीय अनुपात एवम सर्वसमिकाए ( Trigonometry Ratios and Identities ) Exercise 8.4 (Q21-Q30)

Exercise 8.4

Question 21

$\left(\sin ^{8} \theta-\cos ^{8} \theta\right)=\left(\sin ^{2} \theta-\cos ^{2} \theta\right)\left(1-2 \sin ^{2} \theta \cdot \cos ^{2} \theta\right)$

Sol :

LHS

$\left(\sin ^{8} \theta-\cos ^{8} \theta\right)$

$\left[\left(\sin ^{2} \theta\right)^{2}\right]^{2}-\left[\left(\cos ^{2} \theta\right)^{2}\right]^{2}$

$\left[\left(\sin ^{2} \theta\right)^{2}-\left(\cos ^{2} \theta\right)^{2}\right] \times\left[\left(\sin ^{2} \theta\right)^{2}+\left(\cos ^{2} \theta\right)^{2}\right]$

$\left[\left(\sin ^{2} \theta-\cos ^{2} \theta\right)\left(\sin ^{2} \theta+\cos ^{2} \theta\right)\right]\left[\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-\right.\left.2 \sin ^{2} \theta \times \cos ^{2} \theta\right]$

$\left(\sin ^{2} \theta-\cos ^{2} \theta\right) \times 1\left[(1)^{2}-2 \sin ^{2} \theta \times \cos ^{2} \theta\right]$

$\left(\sin ^{2} \theta-\cos ^{2} \theta\right) \times\left(1-2 \sin ^{2} \theta \times \cos ^{2} \theta\right)$

RHS

Question 22

$2\left(\sin ^{6} \theta+\cos ^{6} \theta\right)-3\left(\sin ^{4} \theta+\cos ^{4} \theta\right)+1=0$

Sol :

LHS

$=2\left(\sin ^{6} \theta+\cos ^{6} \theta\right)-3\left(\sin ^{4} \theta+\cos ^{4} \theta\right)+1$

$=\left[\left(\sin ^{2} \theta\right)^{3}+\left(\cos ^{2} \theta\right)^{3}\right]-3\left[\left(\sin ^{2} \theta\right)^{2}+\left(\cos ^{2} \theta\right)^{2}\right]+1$

$=2\left[\left(\sin ^{2} \theta+\cos ^{2} \theta\right)\left(\sin ^{2} \theta\right)^{2}-\sin ^{2} \theta \times \cos ^{2} \theta+\left(\cos ^{2} \theta\right)^{2}\right]-3\left[\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \times \cos ^{2} \theta\right]+1$

$=2 \times 1\left[\left(\sin ^{2} \theta\right)^{2}+\left(\cos ^{2} \theta\right)^{2}-\sin ^{2} \theta \times \cos ^{2} \theta\right]-3 \times\left[(1)^{2}-2 \sin ^{2} \theta \times \cos ^{2} \theta\right]+1$

$=2\left[\left(\sin ^{2} \theta\right)^{2}+\left(\cos ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \times \cos ^{2} \theta\right]-3\left[(1)^{2}-2 \sin ^{2} \theta \times \cos ^{2} \theta\right]+1$

$=2\left[\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \times \cos ^{2} \theta-\sin ^{2} \theta \times \cos ^{2} \theta\right]-3\left[1-2 \sin ^{2} \theta \times \cos ^{2} \theta\right]+1$

$=2\left[(1)^{2}-2 \sin ^{2} \theta \times \cos ^{2} \theta-\sin ^{2} \theta \times \cos ^{2} \theta\right]-3\left[1-2 \sin ^{2} \theta \times \cos ^{2} \theta\right]+1$

$=2\left[1-3 \sin ^{2} \theta \times \cos ^{2} \theta\right]-3\left[1-2 \sin ^{2} \theta \times \cos ^{2} \theta\right]+1$

$=2-6 \sin ^{2} \theta \times \cos ^{2} \theta-3+6 \sin ^{2} \theta \times \cos ^{2} \theta+1$

=2-3+1=3-3=0

Type-III : त्रिकोणमितीय व्यंजकों के वर्गमूल से सम्बद्ध सर्वसमिकाओं को सिद्ध करने पर आधारित प्रश्न :

निम्नलिखित सर्वसमिकाओं को सिद्ध करें :

Question 23

$\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\sin A+\cos A$

Sol :

LHS

$\frac{\cos \mathrm{A}}{1-\tan \mathrm{A}}+\frac{\sin \mathrm{A}}{1-\cot \mathrm{A}}$

$\frac{\cos \mathrm{A}}{1-\frac{\sin \mathrm{A}}{\cos \mathrm{A}}}+\frac{\sin \mathrm{A}}{1-\frac{\cos \mathrm{A}}{\sin \mathrm{A}}}$

$\frac{\cos \mathrm{A}}{\frac{\cos \mathrm{A}-\sin \mathrm{A}}{\cos \mathrm{A}}}+\frac{\sin \mathrm{A}}{\frac{\sin \mathrm{A}-\cos \mathrm{A}}{\sin \mathrm{A}}}$

$\frac{\cos ^{2} \mathrm{~A}}{\cos \mathrm{A}-\sin \mathrm{A}}+\frac{\sin ^{2} \mathrm{~A}}{\sin \mathrm{A}-\cos \mathrm{A}}$

$\frac{\cos ^{2} \mathrm{~A}}{\cos \mathrm{A}-\sin \mathrm{A}}-\frac{\sin ^{2} \mathrm{~A}}{\sin \mathrm{A}-\cos \mathrm{A}}$

$\frac{\cos ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~A}}{\cos \mathrm{A}-\sin \mathrm{A}}$

$\frac{(\cos \mathrm{A}-\sin \mathrm{A})(\cos \mathrm{A}+\sin \mathrm{A})}{\cos \mathrm{A}-\sin \mathrm{A}}$

cos A+sin A 

RHS

Question 24

$\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}=\frac{2}{\sin \theta}$

Sol :

LHS

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

परिमेयकरण करने पर

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

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

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

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

$\frac{2}{\sin \theta}$ 

RHS

Question 25

$\frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}=2 \sec ^{2} \theta$

Sol :

LHS

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

$\frac{1-\sin \theta+1+\sin \theta}{(1+\sin \theta)(1-\sin \theta)}$

$\frac{2}{1^{2}-\sin ^{2} \theta}=\frac{2}{1-\sin ^{2} \theta}$

$\frac{2}{\cos ^{2} \theta}=2 \sec ^{2} \theta$

Proved

Question 26

$\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=2 \sec \theta$

Sol :

LHS

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

परिमेयकरण करने पर

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

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

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

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

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

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

$\frac{2}{\cos \theta}=2 \sec \theta$

RHS

Question 27

$\frac{\cos \theta}{1-\sin \theta}+\frac{\cos \theta}{1+\sin \theta}=\frac{2}{\cos \theta}$

Sol :

LHS

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

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

$\frac{\cos \theta+\cos \theta \times \sin \theta+\cos \theta-\cos \theta \times \sin \theta}{1^{2}-\sin ^{2} \theta}$

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

$\frac{2}{\cos \theta}$ proved

Question 28

$\frac{1}{1+\cos \theta}+\frac{1}{1-\cos \theta}=\frac{2}{\sin ^{2} \theta}$

Sol :

LHS

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

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

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

$\frac{2}{\sin ^{2} \theta}$ proved

Question 29

$\frac{1}{1-\sin \theta}-\frac{1}{1+\sin \theta}=\frac{2 \tan \theta}{\cos \theta}$

Sol :

LHS

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

$\frac{1+\sin \theta-1-\sin \theta}{(1-\sin \theta)(1+\sin \theta)}$

$\frac{1+\sin \theta-1+\sin \theta}{1^{2}-\sin ^{2} \theta}$

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

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

$\frac{2 \tan \theta}{1-\frac{1}{\cos \theta}}$

$\frac{2 \tan \theta}{\cos \theta}$ = proved

Question 30

$\cot ^{2} \theta-\cos ^{2} \theta=\cot ^{2} \theta \cdot \cos ^{2} \theta$

Sol :

LHS

$\cot ^{2} \theta-\cos ^{2} \theta$

$\frac{\cos ^{2} \theta}{\sin ^{2} \theta}-\cos ^{2} \theta$

$\cos ^{2} \theta\left[\frac{1}{\sin ^{2} \theta}-1\right]$

$\cos ^{2} \theta\left[\frac{1-\sin ^{2} \theta}{\sin ^{2} \theta}\right]$

$\cos ^{2} \theta\left[\frac{\cos ^{2} \theta}{\sin ^{2} \theta}\right]$

$\cos ^{2} \theta \times \cot ^{2} \theta$ proved