Simplify cot²θ(1 + tan²θ)

Last Updated : 03 Sep, 2021

The word trigonon means triangle and metron meaning measure. So, trigonometry is the branch of mathematics that deals with the sides and angles of a triangle where one of the angles is 90°. Trigonometry finds its applications in various fields such as engineering, image compression, satellite navigation, and architecture.

Trigonometric function, also known as angle function or circular function, is a function of an angle or arc. It is simply expressed in terms of the ratios of pairs of sides of a right-angled triangle. The six commonly used trigonometric functions are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), cosecant (cosec) angles.

$\sin \theta = \frac{P}{B} \newline \cos \theta = \frac{B}{H} \newline \tan \theta = \frac{P}{B} \newline \cot \theta = \frac{B}{P} \newline \sec \theta = \frac{H}{B} \newline \cosec \theta = \frac{B}{P}$

Where P is the perpendicular, B is the base, and H is the hypotenuse.

A trigonometric equation is an equation involving one or more trigonometric ratios of unknown angles. For example, sin²x – 5 cosx = 1/2.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold for all possible values of the variables. In trigonometry, there are a variety of identities that are used to solve a variety of trigonometric problems. They are as follows,

Pythagorean Trigonometric Identities

$\sin^2θ + \cos^2θ = 1 \newline \sec^2θ-\tan^2θ = 1 \newline \cosec^2θ-\cot^2θ = 1$

Reciprocal Trigonometric Identities

$\tan \theta= \frac{1}{\cot \theta} \newline \sec \theta= \frac{1}{\cos \theta} \newline \cosec \theta= \frac{1}{\sin \theta}$

Co-function Identities

$\sin(\frac{\pi}{2}-\theta)=\cos\theta \newline \cos(\frac{\pi}{2}-\theta)=\sin\theta \newline \tan(\frac{\pi}{2}-\theta)=\cot\theta \newline \cot(\frac{\pi}{2}-\theta)=\tan\theta \newline \sec(\frac{\pi}{2}-\theta)=\cosec\theta \newline \cosec(\frac{\pi}{2}-\theta)=\sec\theta$

Complementary Angle Identities

Supplementary Angle Identities

$\newline \sin(\pi-\theta)=\sin\theta \newline \cos(\pi-\theta)=-\cos\theta \newline \tan(\pi-\theta)=-\tan\theta \newline \cot(\pi-\theta)=-\cot\theta \newline \sec(\pi-\theta)=-\sec\theta \newline \cosec(\pi-\theta)=\cosec\theta$