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
This is helpful for student .those student they are weak in math then they use these sites
ReplyDelete