Simplify cot²θ(1 + tan²θ)

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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

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

Simplify cot²θ(1 + tan²θ)

Solution:

cot²θ(1 + tan²θ)

(1 + tan²θ) = sec²θ

Substituting the value of 1 + tan²θ in the above expression,

= cot²θ × (sec²θ)

Recognize that, $\cot \theta=\frac{\cos \theta}{\sin\theta}$ and $\sec\theta=\frac{1}{\cos\theta}$

On substituting the value of cotθ and secθ in the above expression,

= $\frac{\cos^2 \theta}{\sin^2 \theta} * \frac{1}{\cos^2 \theta}$

= $\frac{1}{\sin^2 \theta} = \cosec^2 \theta$

Sample Questions

Question 1: Find the value of $\mathbf{\tan^2 \theta* (1 + \cot^2 \theta)}$

Solution:

$\mathbf{\tan^2 \theta* (1 + \cot^2 \theta)}$

$1+\cot^2\theta=\cosec^2\theta$

Substituting the value of $1+\cot^2\theta$ in the above expression,

= $\tan^2 \theta*(\cosec^2\theta)$

= $\frac{\sin^2 \theta}{\cos^2 \theta} * \frac{1}{\sin^2 \theta}$

= $\frac{1}{\cos^2 \theta} = \sec^2 \theta$

Question 2: Find the value of $\mathbf{\cot^2 \theta* (1 - \cos^2 \theta)}$

Solution:

$\cot^2 \theta* (1 - \cos^2 \theta)$

$\sin^2 \theta+\cos^2 \theta=1$ therefore, $1-\cos^2 \theta=\sin^2 \theta$

= $\cot^2 \theta* \sin^2\theta$

= $\frac{\cos^2 \theta}{\sin^2 \theta} * \sin^2 \theta$

= $\cos^2\theta$

Question 3: Find the value of $\mathbf{\cot^2 \theta* (1 + \tan^2 \theta)}$

Solution:

$\cot^2 \theta* (1 + \tan^2 \theta)$

$1 + \tan^2 \theta=\sec^2\theta$

$\cot^2 \theta* \sec^2\theta$

Also, $\cot \theta=\frac{\cos\theta}{\sin \theta}$ and $\sec\theta=\frac{1}{\cos\theta}$

$\frac{\cos^2\theta}{\sin^2\theta}*\frac{1}{\cos^2\theta}$

$\frac{1}{\sin^2\theta}$ = $\cosec^2\theta$

Question 4: Find the value of $\mathbf{\cot^2 \theta* (\sec^2 \theta-1)}$

Solution:

$\cot^2 \theta* (\sec^2 \theta-1)$

$1 + \tan^2 \theta=\sec^2\theta$ , therefore, $\sec^2\theta-1 =\tan^2 \theta$

$\cot^2\theta*\tan^2\theta$

Also, we are aware that $\tan\theta=\frac{1}{\cot\theta}$

$\cot^2\theta*\frac{1}{\cot^2\theta}$ =1

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Last Updated : 03 Sep, 2021

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Simplify cot2θ(1 + tan2θ)

Trigonometric Identities

Simplify cot2θ(1 + tan2θ)

Sample Questions

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Simplify cot²θ(1 + tan²θ)

Simplify cot²θ(1 + tan²θ)