Prove that ${\sin }^2\left(\frac{\mathrm{\pi }}{8}+\frac{A}{2}\right)-{\sin }^2\left(\frac{\mathrm{\pi }}{8}-\frac{A}{2}\right)=\frac{1}{\sqrt{2}}\sin A$.

Important Questions on Trigonometrical Ratios of Multiple and Submultiple of Angles

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 17

Prove that ${\cos }^2\alpha +{\cos }^2\left(\alpha +120°\right)+{\cos }^2\left(\alpha -120°\right)=\frac{3}{2}$

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 18

Prove that ${\cos }^4\frac{\mathrm{\pi }}{8}+{\cos }^4\frac{3\mathrm{\pi }}{8}+{\cos }^4\frac{5\mathrm{\pi }}{8}+{\cos }^4\frac{7\mathrm{\pi }}{8}=\frac{3}{2}$.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 19

Prove that ${\sin }^4\frac{\mathrm{\pi }}{8}+{\sin }^4\frac{3\mathrm{\pi }}{8}+{\sin }^4\frac{5\mathrm{\pi }}{8}+{\sin }^4\frac{7\mathrm{\pi }}{8}=\frac{3}{2}$.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 20

Prove that $\cos 2\theta \cos 2\varphi +{\sin }^2(\theta -\varphi )-{\sin }^2(\theta +\varphi )=\cos (2\theta +2\varphi )$.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 21

Prove that $\left(\tan 4A+\tan 2A\right)\left(1-{\tan }^{2}3A{\tan }^{2}A\right)=2\tan 3A{\sec }^{2}A$.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 22

Prove that: $\left(1+\tan \frac{\text{α}}{2}-\sec \frac{\text{α}}{2}\right)\left(1+\tan \frac{\text{α}}{2}+\sec \frac{\text{α}}{2}\right)=\sin \text{α}·{\sec }^2\frac{\text{α}}{2}$.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 23

Find the proper signs to be applied to the radicals in the following formulae: $2\cos \frac{A}{2}=\pm \sqrt{1-\sin A}\pm \sqrt{1+\sin A}$, when $\frac{A}{2}={278}^{\circ }$.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 8 - Trigonometrical Ratios of Multiple and Submultiple of Angles>EXAMPLES XVIII>Q 24

Find the proper signs to be applied to the radicals in the following formulae: $2\sin \frac{A}{2}=\pm \sqrt{1-\sin A}\pm \sqrt{1+\sin A}$, when $\frac{A}{2}=\frac{19\pi }{11}$.