Exercise 8.3
Question 21
\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ}=1
Sol :
L.H.S
$
\begin{aligned}
&\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ} \\
&\sin ^{2} 40^{\circ}+\cos ^{2}\left(90^{\circ}-50^{\circ}\right) \\
&\sin ^{2} 40^{\circ}+\cos ^{2} 40^{\circ} \\
&=1 \quad \text { R.H.S }
\end{aligned}
$
Question 22
\sin ^{2} 29^{\circ}+\sin ^{2} 61^{\circ}=1
Sol :
L.H.S
$
\begin{aligned}
&\sin ^{2} 29^{\circ}+\sin ^{2} 61^{\circ} \\
&\sin ^{2} 29^{\circ}+\cos ^{2}\left(90^{\circ}-61^{\circ}\right)
\end{aligned}
$
\sin ^{2} 29^{\circ}+\cos ^{2} 29^{\circ}
=1 R.H.S
Question 23
\sin \theta \cdot \cos \left(90^{\circ}-\theta\right)+\cos \theta \sin
\left(90^{\circ}-\theta\right)=1
Sol :
L.H.S
$
\begin{aligned}
&\sin \theta \cdot \cos \left(90^{\circ}-\theta\right)+\cos \theta \cdot \sin \left(90^{\circ}-\theta\right) \\
&\sin \theta \times \sin \theta+\cos \theta \times \cos \theta \\
&\sin ^{2} \theta+\cos ^{2} \theta \\
&=1 \quad \text { R.H.S }
\end{aligned}
$
Question 24
\cos \theta \cdot \cos \left(90^{\circ}-\theta\right)-\sin \theta
\cdot \sin \left(90^{\circ}-\theta\right)=0
Sol :
L.H.S
$
\begin{aligned}
&\cos \theta \cdot \cos \left(90^{\circ}-\theta\right)-\sin \theta \cdot \sin \left(90^{\circ}-\theta\right) \\
&\cos \theta \cdot \sin \theta-\sin \theta \cdot \cos \theta \\
&=0 \quad \text { R.H.S }
\end{aligned}
$
Question 25
\sin 42^{\circ} \cdot \cos 48^{\circ}+\cos 42^{\circ} \cdot \sin
48^{\circ}=1
Sol :
L.H.S
\sin 42^{\circ} \cdot \cos 48^{\circ}+\cos 42^{\circ} \cdot \sin 48^{\circ}
\sin 42^{\circ} \cdot \sin \left(90^{\circ}-48^{\circ}\right)+\cos 42^{\circ} \cdot \cos \left(90^{\circ}-48^{\circ}\right)
\sin 42^{\circ} \cdot \sin 42^{\circ}+\cos 42^{\circ} \cdot \cos 42^{\circ}
=1 R.H.S
Question 26
\frac{\cos 20^{\circ}}{\sin 70^{\circ}}+\frac{\cos \theta}{\sin
\left(90^{\circ}-\theta\right)}=2
Sol :
L.H.S
$
\begin{aligned}
&\frac{\cos 20^{\circ}}{\sin 70^{\circ}}+\frac{\cos \theta}{\sin \left(90^{\circ}-\theta\right)} \\
&\frac{\cos 20^{\circ}}{\sin 70^{\circ}}+\frac{\cos \theta}{\cos \theta)} \\
&\frac{\sin \left(90^{\circ}-20^{\circ}\right)}{\sin 70^{\circ}}+1 \\
&\frac{\sin 70^{\circ}}{\sin 70^{\circ}}+1 \end{aligned}$
1+1=2 R.H.S
Question 27
\sin ^{2} 85^{\circ}+\sin ^{2} 5^{\circ}+\sin ^{2} 6 ?^{\circ}+\sin ^{2},
23^{\circ}=2
Sol :
L.H.S
$
\begin{aligned}
&\sin ^{2} 85^{\circ}+\sin ^{2} 5^{\circ}+\sin ^{2} 67^{\circ}+\sin ^{2} 23^{\circ} \\
&\sin ^{2} 85^{\circ}+\cos ^{2}\left(90^{\circ}-5^{\circ}\right)+\sin ^{2} 67^{\circ}+\cos ^{2}\left(90^{\circ}-23^{\circ}\right) \\
&\sin ^{2} 85^{\circ}+\cos ^{2} 85^{\circ}+\sin ^{2} 67^{\circ}+\cos ^{2} 67^{\circ}
\end{aligned}
$
1+1=2 RHS
Question 28
\tan 9^{\circ} \cdot \tan 27^{\circ} \cdot \tan 45^{\circ} \cdot \tan
63^{\circ} \cdot \tan 81^{\circ}=1
Sol :
L.H.S
\tan 9^{\circ} \cdot \tan 27^{\circ} \cdot \tan 45^{\circ} \cdot \tan 63^{\circ} \cdot \tan 81^{\circ}
\left(\tan 9^{\circ} \cdot \tan 81^{\circ}\right) \cdot\left(\tan 27^{\circ} \cdot \tan 63^{\circ}\right) \cdot \tan 45^{\circ}
\left(\tan 9^{\circ} \cdot \cot \left(90^{\circ}-81^{\circ}\right)\right) \cdot\left(\tan 27^{\circ} \cdot \cot \left(90^{\circ}-63^{\circ}\right)\right) \cdot \times 1
\left(\tan 9^{\circ} \cdot \cot 9^{\circ}\right) \cdot\left(\tan 27^{\circ} \cdot \cot 27^{\circ}\right) \cdot \times 1
=1×1=1 RHS
Question 29
\sin 9^{\circ} \cdot \sin 27^{\circ} \cdot \sin 63^{\circ} \cdot \sin
^{\circ} 81^{\circ}=\cos 9^{\circ} \cdot \cos 27^{\circ} \cdot \cos
63^{\circ} \cdot \cos 81^{\circ}
Sol :
L.H.S
\sin 9^{\circ} \cdot \sin 27^{\circ} \cdot \sin 63^{\circ} \cdot \sin 81^{\circ}
\cos \left(90^{\circ}-9^{\circ}\right) \cdot \cos \left(90^{\circ}-27^{\circ}\right) \cdot \cos \left(90^{\circ}-63^{\circ}\right) \cdot \cos \left(90^{\circ}-81^{\circ}\right)
\cos 81^{\circ} \cdot \cos 63^{\circ} \cdot \cos 27^{\circ} \cdot \cos 9^{\circ}
\cos 9^{\circ} \cdot \cos 27^{\circ} \cdot \cos 63^{\circ} \cdot \cos 81^{\circ}
R.H.S
Question 30
(i) \tan 7^{\circ} \cdot \tan 23^{\circ}, \tan 60^{\circ} \cdot \tan
67^{\circ} \cdot \tan 83^{\circ}=\sqrt{3}
Sol :
L.H.S
\tan 7^{\circ} \cdot \tan 23^{\circ} \cdot \tan 60^{\circ} \cdot \tan 67^{\circ} \cdot \tan 83^{\circ}
\left(\tan 7^{\circ} \cdot \tan 83^{\circ}\right) \cdot\left(\tan 23^{\circ} \cdot \tan 67^{\circ}\right) \cdot \tan 60^{\circ}
\left(\tan 7^{\circ} \cdot \cot \left(90^{\circ}-83^{\circ}\right)\right) \cdot\left(\tan 23^{\circ} \cdot \cot \left(90^{\circ}-67^{\circ}\right)\right) \cdot \times \sqrt{3}
\left(\tan 7^{\circ} \cdot \cot 7^{\circ}\right) \cdot\left(\tan 23^{\circ} \cdot \cot 23^{\circ}\right) \cdot \times \sqrt{3}
=1 \times 1 \times \sqrt{3}=\sqrt{3} R.H.S
(ii) \tan 15^{\circ} \tan 25^{\circ} \tan 60^{\circ} \tan 65^{\circ} \tan
75^{\circ}=\sqrt{3}
Sol :
L.H.S
\tan 15^{\circ} \cdot \tan 25^{\circ} \cdot \tan 60^{\circ} \cdot \tan 65^{\circ} \cdot \tan 75^{\circ}
\left(\tan 15^{\circ} \cdot \tan 75^{\circ}\right) \cdot\left(\tan 25^{\circ} \cdot \tan 65^{\circ}\right) \cdot \tan 60^{\circ}
\left(\tan 15^{\circ} \cdot \cot \left(90^{\circ}-75^{\circ}\right)\right) \cdot\left(\tan 25^{\circ} \cdot \cot \left(90^{\circ}-65^{\circ}\right)\right) \cdot \times \sqrt{3}
\left(\tan 15^{\circ} \cdot \cot 15^{\circ}\right) \cdot\left(\tan 25^{\circ} \cdot \cot 25^{\circ}\right) \cdot \times \sqrt{3}
=1 \times 1 \times \sqrt{3}=\sqrt{3} R.H.S
(iii) \frac{2 \sin ^{2} 63^{\circ}+1+2 \sin ^{2} 27^{\circ}}{3 \cos ^{2}
17^{\circ}-2+3 \cos ^{2} 73^{\circ}}=3
Sol :
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