8.2 Q-7

Exercise 8.2 — Question 7

Topic: Double-Angle and Half-Angle Identities

This question involves applying double-angle and half-angle identities derived from the fundamental law of trigonometry.

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Key Identities

Double-Angle Identities:

$\sin 2\theta =2\sin \theta \cos \theta$

$\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta =2{\cos }^2\theta -1=1-2{\sin }^2\theta$

$\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$

Half-Angle Identities:

$\sin \frac{\theta }{2}=\pm \sqrt{\frac{1-\cos \theta }{2}}$

$\cos \frac{\theta }{2}=\pm \sqrt{\frac{1+\cos \theta }{2}}$

$\tan \frac{\theta }{2}=\pm \sqrt{\frac{1-\cos \theta }{1+\cos \theta }}$

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Worked Example: Verify ${\cos }^{4}x-{\sin }^{4}x=\cos 2x$

Step 1: Factor the left-hand side as a difference of squares: ${\cos }^{4}x-{\sin }^{4}x=({\cos }^{2}x-{\sin }^{2}x)({\cos }^{2}x+{\sin }^{2}x)$

Step 2: Apply the Pythagorean identity ${\cos }^{2}x+{\sin }^{2}x=1$: $({\cos }^{2}x-{\sin }^{2}x)(1)={\cos }^{2}x-{\sin }^{2}x$

Step 3: Apply the double-angle identity: ${\cos }^{2}x-{\sin }^{2}x=\cos 2x✓$

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