Exercise 8.3
Question 41
$\left(\frac{\sin 35^{\circ}}{\cos 55^{\circ}}\right)^{2}+\left(\frac{\cos
55^{\circ}}{\sin 35^{\circ}}\right)-2 \cos 60^{\circ}$
Sol :
$\frac{\cos \left(90^{\circ}-3^{\circ}\right)}{\cos 55^{\circ}}+\frac{\sin \left(90^{\circ}-55^{\circ}\right)}{\sin 35^{\circ}}-2 \times \frac{1}{2}$
$\frac{\cos 55^{\circ}}{\cos 55^{\circ}}+\frac{\sin 35^{\circ}}{\sin 35^{\circ}}-1$
1+1-1=2-1=1
Question 42
$\frac{\cos 80^{\circ}}{\sin 10^{\circ}}+\cos 59^{\circ} \cdot
\operatorname{cosec} 31^{\circ}$
Sol :
$\cos \left(90^{\circ}-\left(50^{\circ}+\theta\right)\right)-\cos \left(40^{\circ}-\theta\right)+\tan 1^{\circ} \cdot \tan 89^{\circ} \cdot \tan 10^{\circ} \cdot \tan 80^{\circ} \cdot \tan 20^{\circ} \cdot \tan 70^{\circ}$
$\cos \left(90^{\circ}-50^{\circ}-\theta\right)-\cos \left(40^{\circ}-\theta\right)+\tan 1^{\circ} \cdot \cot \left(90^{\circ}-89^{\circ}\right) \cdot \tan 10^{\circ} \cdot \cot \left(90^{\circ}-80^{\circ}\right) \cdot \tan 20^{\circ} \cdot \cot \left(90^{\circ}-70^{\circ}\right)$
$\cos \left(40^{\circ}-\theta\right)-\cos \left(40^{\circ}-\theta\right)+\tan 1^{\circ} \cdot \cot 1^{\circ} \cdot \tan 10^{\circ} \cdot \cot 10^{\circ} \cdot \tan 20^{\circ} \cdot \cot 20^{\circ}$
=0+1×1×1=1
Question 43
$\sin \left(50^{\circ}+\theta\right)-\cos
\left(40^{\circ}-\theta\right)+\tan 1^{\circ} \cdot \tan 10^{\circ} \cdot
\tan 20^{\circ} \cdot \tan 70^{\circ} \cdot \tan 80^{\circ} \tan 89^{\circ}$
Sol :
$\operatorname{cosec}^{2}\left(90^{\circ}-10^{\circ}\right)-\cot ^{2} 80^{\circ}+\frac{\sin 15^{\circ} \times \sin \left(90^{\circ}-75^{\circ}\right)+\cos 15^{\circ} \times \cos \left(90^{\circ}-75^{\circ}\right)}{\cos \theta \times \cos \theta+\sin \theta \times \sin \theta}$
$\operatorname{cosec}^{2} 80^{\circ}-\cot ^{2} 80^{\circ}+\frac{\sin 15^{\circ} \times \sin 15^{\circ}+\cos 15^{\circ} \times \cos 15^{\circ}}{\cos ^{2} \theta+\sin ^{2} \theta}$
$1+\frac{\sin ^{2} 15^{\circ}+\cos ^{2} 15^{\circ}}{\cos ^{2} \theta+\sin ^{2} \theta}$
$1+\frac{1}{1}=1+1=2$
Question 44
$\sec ^{2} 10^{\circ}-\cot ^{2} 80^{\circ}+\frac{\sin 15^{\circ} \cdot \cos
75^{\circ}+\cos 15^{\circ} \cdot \sin 75^{\circ}}{\cos \theta \sin
\left(90^{\circ}-\theta\right)+\sin \theta \cos
\left(90^{\circ}-\theta\right)}$
Sol :
$\sin \left(90^{\circ}-\left(90^{\circ}-\left(40^{\circ}+\theta\right)\right)-\sin \left(50^{\circ}-\theta\right)+\frac{\sin ^{2}\left(90^{\circ}-40^{\circ}\right)+\cos ^{2} 50^{\circ}}{\cos ^{2}\left(90^{\circ}-40^{\circ}\right)+\sin ^{2} 50^{\circ}}\right.$
$\sin \left(90^{\circ}-40^{\circ}-\theta\right)-\sin \left(50^{\circ}-\theta\right)+\frac{\sin ^{2} 50^{\circ}+\cos ^{2} 50^{\circ}}{\cos ^{2} 50^{\circ}+\sin ^{2} 50^{\circ}}$
$\sin \left(50^{\circ}-\theta\right)-\left(50^{\circ}-\theta\right)+1$
$0+\frac{1}{1}=1$
Question 45
$\cos \left(40^{\circ}+\theta\right)-\sin
\left(50^{\circ}-\theta\right)+\frac{\cos ^{2} 40^{\circ}+\cos ^{2}
50^{\circ}}{\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ}}$
Sol :
$\sin \left(90^{\circ}-\left(90^{\circ}-\left(40^{\circ}+\theta\right)\right)-\sin \left(50^{\circ}-\theta\right)+\frac{\sin ^{2}\left(90^{\circ}-40^{\circ}\right)+\cos ^{2} 50^{\circ}}{\cos ^{2}\left(90^{\circ}-40^{\circ}\right)+\sin ^{2} 50^{\circ}}\right.$
$\sin \left(90^{\circ}-40^{\circ}-\theta\right)-\sin \left(50^{\circ}-\theta\right)+\frac{\sin ^{2} 50^{\circ}+\cos ^{2} 50^{\circ}}{\cos ^{2} 50^{\circ}+\sin ^{2} 50^{\circ}}$
$\sin \left(50^{\circ}-\theta\right)-\left(50^{\circ}-\theta\right)+1$
$0+\frac{1}{1}=1$
Question 46
$\frac{\cos 70^{\circ}}{\sin 20^{\circ}}+\frac{\cos 55^{\circ} \cdot
\operatorname{cosec} 35^{\circ}}{\tan 5^{\circ} \cdot \tan 25^{\circ} \cdot
\tan 45^{\circ} \cdot \tan 65^{\circ} \cdot \tan 85^{\circ}}$
Sol :
$\frac{\sin \left(90^{\circ}-70^{\circ}\right)}{\sin 20^{\circ}}+\frac{\sin \left(90^{\circ}-55^{\circ}\right) \times \operatorname{cosec} 35^{\circ}}{\left(\tan 5^{\circ} \times \tan 85^{\circ}\right) \times\left(\tan 25^{\circ} \times \tan 65^{\circ}\right) \times \tan 45^{\circ}}$
$\frac{\sin 20^{\circ}}{\sin 20^{\circ}}+\frac{\sin 35^{\circ} \times \operatorname{cosec} 35^{\circ}}{\left(\tan 5^{\circ} \times \cot \left(90^{\circ}-85^{\circ}\right) \times\left(\tan 25^{\circ} \times \cot \left(90^{\circ}-65^{\circ}\right) \tan 45^{\circ}\right.\right.}$
$1+\frac{1}{\left(\tan 5^{\circ} \times \cot 5^{\circ}\right)\left(\tan 25^{\circ} \times \cot 25^{\circ}\right) \times 1}$
$1+\frac{1}{1 \times 1 \times 1}=1+\frac{1}{1}=1+1=2$
Question 47
$\left(\frac{\sin 27^{\circ}}{\cos 63^{\circ}}\right)^{2}+\left(\frac{\cos
63^{\circ}}{\sin 27^{\circ}}\right)^{2}$
Sol :
$\left(\frac{\cos \left(90^{\circ}-27^{\circ}\right)}{\cos 63^{\circ}}\right)^{2}+\left(\frac{\sin \left(90^{\circ}-63^{\circ}\right)}{\sin 27^{\circ}}\right)^{2}$
$\left(\frac{\cos 63^{\circ}}{\cos 63^{\circ}}\right)^{2}+\left(\frac{\sin 27^{\circ}}{\sin 27^{\circ}}\right)^{2}$
1+1=2
Question 48
निम्नलिखित के मान ज्ञात कीजिए ।
(i) $\frac{3 \sin 5^{\circ}}{\cos 85^{\circ}}+\frac{2 \cos
33^{\circ}}{\sin 57^{\circ}}$
Sol :
$\frac{3 \cos \left(90^{\circ}-5^{\circ}\right)}{\cos 85^{\circ}}+\frac{2 \sin \left(90^{\circ}-33^{\circ}\right)}{\sin 57^{\circ}}$
$\frac{3 \cos 65^{\circ}}{\cos 85^{\circ}}+\frac{2 \sin 57^{\circ}}{\sin 57^{\circ}}$
3×1+2×1=3+2
=5
(ii) $\frac{\cot 54^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\cot
70^{\circ}}-2$
Sol :
$\frac{\tan \left(90^{\circ}-54^{\circ}\right)}{\tan 36^{\circ}}+\frac{\cot \left(90^{\circ}-20^{\circ}\right)}{\cot 70^{\circ}}-2$
$\frac{\tan 36^{\circ}}{\tan 36^{\circ}}+\frac{\cot 76^{\circ}}{\cot 70^{\circ}}-2$
=1+1-2=0
(iii) $\frac{\cos 80^{\circ}}{\sin 10^{\circ}}+\cos 59^{\circ}
\operatorname{cosec} 31^{\circ}$
Sol :
$\frac{\sin \left(90^{\circ}-80^{\circ}\right)}{\sin 10^{\circ}}+\sin \left(90^{\circ}-59^{\circ}\right) \times \operatorname{cosec} 31^{\circ}$
$\frac{\sin 30^{\circ}}{\sin 10^{\circ}}+\sin 31^{\circ} \times \operatorname{cosec} 31^{\circ}$
1+1=2
(iv) $\cos 38^{\circ} \cos 52^{\circ}-\sin 38^{\circ} \sin 52^{\circ}$
Sol :
$\sin \left(90^{\circ}-38^{\circ}\right) \times \cos 52^{\circ}-\cos \left(90^{\circ}-38^{\circ}\right) \times \sin 52^{\circ}$
$\sin 52^{\circ} \times \cos 52^{\circ}-\cos 52^{\circ} \times \sin 52^{\circ}$
1-1=0
(v) $\sec 41^{\circ} \sin 49^{\circ}+\operatorname{coss} 49^{\circ}
\operatorname{cosec} 41^{\circ}$
Sol :
$\operatorname{cosec}\left(90^{\circ}-41^{\circ}\right) \times \sin 49^{\circ}+\sin \left(90^{\circ}-49^{\circ}\right) \times \operatorname{cosec} 41^{\circ}$
$\operatorname{cosec} 49^{\circ} \times \sin 49^{\circ}+\sin 41^{\circ} \times \operatorname{cosec} 41^{\circ}$
1+1=2
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