M L Aggarwal Solutions for Exercise 3: Chapter Test

Evaluate the following:

$2\left(\frac{{\cos }^2 20°+{\cos }^2 70°}{{\sin }^2 25°+{\sin }^2 65°}\right)-\tan 45°+\tan 13°\tan 23°\tan 30°\tan 67°\tan 77°$

$\frac{{\sin }^2 22°+{\sin }^2 68°}{{\cos }^2 22°+{\cos }^2 68°}+{\sin }^2 63°+\cos 63° \sin 27°$

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined: ${\sec }^4 A\left(1-{\sin }^4 A\right)-2 {\tan }^2 A=1$

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined: ${\left(\sec A-\tan A\right)}^2\left(1+\sin A\right)=1-\sin A$

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined: $\left(\sec A-cosec A\right)\left(1+\tan A+\cot A\right)=\tan A \sec A-\cot A cosec A$

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined: $\frac{{\tan }^{2 }\theta }{{\tan }^2 \theta -1}+\frac{{cosec}^{2 }\theta }{{\sec }^2 \theta -{cosec}^2 \theta }=\frac{1}{{\sin }^2 \theta -{\cos }^2 \theta }$

When $0°<\theta <90°$, solve the following equation: $2co{s}^2 \theta +sin \theta -2=0$

When $0°<\theta <90°$, solve the following equations: $3 \cos \theta =2 {\sin }^2 \theta$