5.5.3 Trigonometry - Further Identities

In trigonometry, further identities extend beyond the basic identities like the Pythagorean identities, reciprocal identities, and quotient identities. These additional identities are useful in simplifying expressions, solving equations, and analysing trigonometric functions. Below are some of the most important further identities:

1. Sum and Difference Identities:

These identities allow you to find the sine, cosine, or tangent of the sum or difference of two angles.

• Sine of a Sum/Difference: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
  • Example: $\sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$
• Cosine of a Sum/Difference: $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
  • Example: $\cos(15^\circ) = \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ$
• Tangent of a Sum/Difference: $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
  • Example: $\tan(75^\circ) = \tan(45^\circ + 30^\circ) = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ}$

2. Double Angle Identities:

These identities express trigonometric functions of double angles ($2$ $\theta$) in terms of single angles.

• Sine of a Double Angle: $\sin 2\theta = 2\sin \theta \cos \theta$
• Cosine of a Double Angle: $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$ This identity can also be written as: $\cos 2\theta = 2\cos^2 \theta - 1$ or $\cos 2\theta = 1 - 2\sin^2 \theta$
• Tangent of a Double Angle: $\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$

3. Half-Angle Identities:

These identities express trigonometric functions of half angles ($\frac{\theta}{2}$) in terms of the full angle.

• Sine of a Half Angle: $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$ The sign $\pm$ depends on the quadrant in which$\frac{\theta}{2}$ lies.
• Cosine of a Half Angle: $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$ Again, the sign$\pm$ depends on the quadrant.
• Tangent of a Half Angle: $\tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{\sin \theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta}$

4. Product-to-Sum Identities:

These identities convert products of sine and cosine into sums, making them easier to integrate or simplify.

• Product of Sines: $\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]$
• Product of Cosines: $\cos A \cos B = \frac{1}{2} [\cos(A - B) + \cos(A + B)]$
• Product of Sine and Cosine: $\sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)]$

5. Sum-to-Product Identities:

These identities convert sums or differences of sines and cosines into products, which can be useful in solving equations.

• Sum of Sines: $\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)$
• Difference of Sines: $\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)$
• Sum of Cosines: $\cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)$
• Difference of Cosines: $\cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)$

6. Co-Function Identities:

These identities show the relationship between trigonometric functions of complementary angles.

• Sine and Cosine: $\sin(90^\circ - \theta) = \cos \theta$ $\cos(90^\circ - \theta) = \sin \theta$
• Tangent and Cotangent: $\tan(90^\circ - \theta) = \cot \theta$ $\cot(90^\circ - \theta) = \tan \theta$
• Secant and Cosecant: $\sec(90^\circ - \theta) = \csc \theta$ $\csc(90^\circ - \theta) = \sec \theta$

7. Even-Odd Identities:

These identities describe how trigonometric functions behave when their input is negated.

• Sine and Cosecant (Odd Functions): $\sin(-\theta) = -\sin \theta$ $\csc(-\theta) = -\csc \theta$
• Cosine and Secant (Even Functions): $\cos(-\theta) = \cos \theta$ $\sec(-\theta) = \sec \theta$
• Tangent and Cotangent (Odd Functions): $\tan(-\theta) = -\tan \theta$ $\cot(-\theta) = -\cot \theta$

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Now let's look at some examples:

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Summary: