CBSE Class 10 Chapter 8 Introduction of Trigonometry - Trigonometric Identities

 Trigonometric Identities of Opposite Angles

• \(\sin(-\theta) = -\sin \theta\)
• \(\cos(-\theta) = \cos \theta\)
• \(\tan(-\theta) = -\tan \theta\)
• \(\cot(-\theta) = -\cot \theta\)
• \(\sec(-\theta) = \sec \theta\)
• \(\csc(-\theta) = -\csc \theta\)

Trigonometric Identities of Complementary Angles

When the sum of two angle are equal to \(90^{0}\) , they are known as complementary angles.

• \(\sin(90 - \theta) = \cos \theta\)

Proof:

Using the sum-to-product identity for sine:

\( \sin(90 - \theta) = \sin 90 \cos \theta - \cos 90 \sin \theta \)

Simplifying with \(\sin 90 = 1\) and \(\cos 90 = 0\):

\( \sin(90 - \theta) = \cos \theta \)

Therefore, \(\sin (90-\theta)=\cos\theta\) is proven.

• \(\cos(90 - \theta) = \sin \theta\)

Proof:

Using the sum-to-product identity for cosine:

\( \cos(90 - \theta) = \cos 90 \cos \theta + \sin 90 \sin \theta \)

Simplifying with \(\cos 90 = 0\) and \(\sin 90 = 1\):

\( \cos(90 - \theta) = \sin \theta \)

Therefore, \(\cos(90 - \theta) = \sin \theta\) is proven.

• \(\tan(90 - \theta) = \cot \theta\)

Proof:

Using the relationship \(\tan\theta=\frac {\sin\theta}{\cos\theta}\):

We know that  \(\sin(90 - \theta) = \cos \theta\) and \(\cos(90 - \theta) = \sin \theta\)

\( \tan(90 - \theta) = \frac{\sin(90 - \theta)}{\cos(90 - \theta)} \)

Therefore, \(\tan(90 - \theta) = \cot \theta\) is proven.

• \(\cot(90 - \theta) = \tan \theta\)

Proof:

Using the relationship \(cot\theta = \frac {1}{\tan\theta}\):

We know that  \(\tan(90 - \theta) = \cot \theta\) ;

\( \cot(90 - \theta) = \frac{1}{\cot \theta} = \tan \theta \)

Therefore, \(\cot(90 - \theta) = \tan \theta\) is proven.

\(\sec(90 - \theta) = \csc \theta\)

\(\csc(90 - \theta) = \sec \theta\)

Trigonometric Identities of Supplementary Angles

Two angles are supplementary if their sum is equal to 90 degrees. Similarly, when we can learn here the trigonometric identities for supplementary angles.

\(\sin(180^\circ - \theta) = \sin \theta\)

\(\cos(180^\circ - \theta) = -\cos \theta\)

\(\csc(180^\circ - \theta) = \csc \theta\)

\(\sec(180^\circ - \theta) = -\sec \theta\)

\(\tan(180^\circ - \theta) = -\tan \theta\)

\(\cot(180^\circ - \theta) = -\cot \theta\)

Sum and Difference of Angles Trigonometric Identities

Consider two angles , α and β, the trigonometric sum and difference identities are as follows:

\(\sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)\)

\(\sin(\alpha - \beta) = \sin(\alpha) \cos(\beta) - \cos(\alpha) \sin(\beta)\)

\(\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)\)

\(\cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta)\)

\(\tan (\alpha+\beta)=\frac {\tan\alpha+\tan\beta}{1-\tan\alpha.\tan\beta}\)

\(\tan (\alpha-\beta)=\frac {\tan\alpha-\tan\beta}{1+\tan\alpha.\tan\beta}\)

Double Angle Trigonometric Identities

If the angles are doubled, then the trigonometric identities for sin, cos and tan are:

\(\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\)

\(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2 \cos^2(\theta) - 1 = 1 - 2 \sin^2(\theta)\)

\(\tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)}\)

Half Angle Identities

If the angles are halved, then the trigonometric identities for sin, cos and tan are:

\(\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{2}}\)

\(\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}}\)

\(\tan\left(\frac{\theta}{2}\right) = \pm \sqrt{(1 - \cos\theta)(1 + \cos\theta)}\)

Product-Sum Trigonometric Identities

The product-sum trigonometric identities change the sum or difference of sines or cosines into a product of sines and cosines.

\(\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)\)

\(\cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)\)

\(\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)\)

\(\cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)\)

Trigonometric Identities of Products

These identities are:

\(\sin A \cdot \sin B = \frac{\cos(A - B) - \cos(A + B)}{2}\)

\(\sin A \cdot \cos B = \frac{\sin(A + B) + \sin(A - B)}{2}\)

\(\cos A \cdot \cos B = \frac{\cos(A + B) + \cos(A - B)}{2}\)