Chapter 11

$$ \cos ^{4} A-\sin ^{4} A+1=2 \cos ^{2} A $$

$$ (\sin A+\cos A)(1-\sin A \cos A)=\sin ^{3} A+\cos ^{3} A $$

$$ \frac{\sin A}{1+\cos A}+\frac{1+\cos A}{\sin A}=2 \operatorname{cosec} A $$

$$ \cos ^{\circ} A+\sin ^{\circ} A=1-3 \sin ^{2} A \cos ^{2} A $$

$$ \sqrt{\frac{1-\sin A}{1+\sin A}}=\sec A-\tan A $$

$$ \frac{\operatorname{cosec} A}{\operatorname{cosec} A-1}+\frac{\cos e c A}{\operatorname{cosec} A+1}=2 \sec ^{2} A $$

$$ \frac{\operatorname{cosec} A}{\cot A+\tan A}=\cos A $$

$$ (\sec A+\cos A)(\sec A-\cos A)=\tan ^{2} A+\sin ^{2} A $$

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

$$ \frac{1}{\sec A-\tan A}=\sec A+\tan A $$

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

$$ \frac{1+\tan ^{2} A}{1+\cot ^{2} A}=\frac{\sin ^{2} A}{\cos ^{2} A} $$

$$ \frac{\sec A-\tan A}{\sec A+\tan A}=1-2 \sec A \tan A+2 \tan ^{2} A $$

$$ \frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}=\sec A \operatorname{cosec} A+1 $$

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

$$ (\sin A+\cos A)(\cot A+\tan A)=\sec A+\operatorname{cosec} A $$

$$ \sec ^{4} A-\sec ^{2} A=\tan ^{4} A+\tan ^{2} A $$

$$ \cot ^{4} A+\cot ^{2} A=\operatorname{cosec}^{4} A-\operatorname{cosec}^{2} A $$

$$ \sqrt{\operatorname{cosec}^{2} A-1}=\cos A \operatorname{cosec} A $$

$$ \sec ^{2} A \operatorname{cosec}^{2} A=\tan ^{2} A+\cot ^{2} A+2 $$

$$ \tan ^{2} A-\sin ^{2} A=\sin ^{4} A \sec ^{2} A $$

$$ (1+\cot A-\operatorname{cosec} A)(1+\tan A+\sec A)=2 $$

$$ \frac{1}{\cos e c A-\cot A}-\frac{1}{\sin A}=\frac{1}{\sin A}-\frac{1}{\operatorname{cosec} A+\cot A} $$

$$ \frac{\cot A \cos A}{\cot A+\cos A}=\frac{\cot A-\cos A}{\cot A \cos A} $$

$$ \frac{\cot A+\tan B}{\cot B+\tan A}=\cot A \tan B $$

$$ \left(\frac{1}{\sec ^{2} A-\cos ^{2} A}+\frac{1}{\operatorname{cosec}^{2} A-\sin ^{2} A}\right) \cos ^{2} A \sin ^{2} A=\frac{1-\cos ^{2} A \sin ^{2} A}{2+\cos ^{2} A \sin ^{2} A} $$

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

$$ \frac{\cos A \operatorname{cosec} A-\sin A \sec A}{\cos A+\sin A}=\operatorname{cosec} A-\sec A $$

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

$$ (\tan A+\operatorname{cosec} B)^{2}-(\cot B-\sec A)^{2}=2 \tan A \cot B(\operatorname{cosec} A+\sec B) $$

$$ 2 \sec ^{2} A-\sec ^{4} A-2 \operatorname{cosec}^{2} A+\operatorname{cosec}^{4} A=\cot ^{4} A-\tan ^{4} A $$

$$ (\sin A+\operatorname{cosec} A)^{2}+(\cos A+\sec A)^{2}=\tan ^{2} A+\cot ^{2} A+7 $$

$$ (1+\cot A+\tan A)(\sin A-\cos A)=\frac{\sec A}{\operatorname{cosec}^{2} A}-\frac{\cos e c A}{\sec ^{2} A} $$

$$ \text { If the angle } \alpha \text { is in the third quadrant and } \tan \alpha=2 \text { , then find the value of } \sin \alpha \text { . } $$

$$ \text { If } \theta \text { is an acute angle and } \tan \theta=\frac{1}{\sqrt{7}}, \text { then find the value of } \frac{\cos e c^{2} \theta-\sec ^{2} \theta}{\cos e c^{2} \theta+\sec ^{2} \theta} \text { . } $$

$$ \text { If } \tan \theta=\frac{p}{q}, \text { show that } \frac{p \sin \theta-q \cos \theta}{p \sin \theta+q \cos \theta}=\frac{p^{2}-q^{2}}{p^{2}+q^{2}} \text { . } $$

$$ \text { If } \tan ^{2} \theta=1-a^{2}, \text { prove that } \sec \theta+\tan ^{3} \theta \operatorname{cosec} \theta=\left(2-a^{2}\right)^{\frac{3}{2}} \text { . } $$

$$ \text { If } \sec x=p+\frac{1}{4 p}, \text { show that } \sec x+\tan x=2 p \text { or } \frac{1}{2 p} \text { . } $$

$$ \sin (\pi+\theta) \sin (\pi-\theta) \operatorname{cosec}^{2} \theta=-1 $$

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

$$ \sin 75^{\circ}+\cos 75^{\circ}=\sqrt{\frac{3}{2}} $$

$$ \sin 105^{\circ}+\cos 105^{\circ}=\cos 45^{\circ} $$

$$ \cos 255^{\circ}+\sin 165^{\circ}=0 $$

$$ \sin 75^{\circ}-\sin 15^{\circ}=\cos 105^{\circ}+\cos 15^{\circ} $$

$$ \sin ^{2} 72^{\circ}-\sin ^{2} 60^{\circ}=\frac{\sqrt{5}-1}{8} $$

$$ \sin 36^{\circ} \sin 72^{\circ} \sin 108^{\circ} \sin 144^{\circ}=\frac{2}{16} $$

$$ \sin \frac{\pi}{10}+\sin \frac{13 \pi}{10}=-\frac{1}{2} $$

$$ \sin \frac{\pi}{10} \sin \frac{13 \pi}{10}=-\frac{1}{4} $$

$$ \sin ^{2} 5^{\circ}+\sin ^{2} 10^{\circ}+\sin ^{2} 15^{\circ}+\ldots \ldots \ldots+\sin ^{2} 90^{\circ}=9 \frac{1}{2} $$

$$ \cos ^{2} 5^{\circ}+\cos ^{2} 10^{\circ}+\cos ^{2} 15^{\circ}+\ldots \ldots \ldots+\cos ^{2} 90^{\circ}=8 \frac{1}{2} $$