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Prove that cos2θ (1+tan2θ) = 1

  • Last Updated : 22 Sep, 2021

Trigonometry is the branch of mathematics that studies the relationships between ratios of the sides of a right-angled triangle with its angles. Although trigonometry does not have direct applications in solving real problems, it is applied in a variety of activities. For example, in music, sound travels in the form of waves, and while this pattern is not as regular as a sine or cosine function, it is nonetheless beneficial in computer music development.

Trigonometric Functions

Trigonometric functions, also known as Circular Functions, are defined as the functions of an angle of a triangle. This means that these trigonometric functions determine the relationship between the angles and sides of a triangle. Sine, cosine, tangent, cotangent, secant, and cosecant are the basic trigonometric functions. 

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Trigonometric Equation 



An equation involving one or more trigonometric ratios of unknown angles is known as a trigonometric equation. It is expressed as ratios of sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), cosecant (cosec) angles. For example, tan2x – 3 = 0

Trigonometric Equations Formulae

There are basic formulae defined for all 6 trigonometric angles with respect to the sides of a right angled triangle. If the angle adjacent to the right angle is known, the ratio of the sides can be easily found out, 

\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.

Trigonometric Identities

Trigonometric identities are equations that relate to different trigonometric functions and are true for any value of the variable that is there in the domain. Below is the relation among trigonometric functions, also known as reciprocal trigonometric identities,

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

Pythagorean Trigonometric Identities



  • sin2θ + cos2θ = 1
  • sec2θ − tan2θ = 1
  • cosec2θ − cot2θ = 1

Prove that cos2θ (1 + tan2 θ) = 1.

Solution:

To prove:  cos2θ (1 + tan2 θ) = 1.

Starting from left hand side

=> cos2θ (1 + tan2 θ) = 1.

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

=> \cos^2 \theta * (\sec^2 \theta)

Also, \sec \theta=\frac{1}{\cos \theta}

=> cos^2 \theta * (\frac{1}{cos^2 \theta})

=> \frac{cos^2 \theta}{cos^2 \theta}=1 = RHS

From here, it is seen that the value of the left hand side will always be equal to 1. Hence, it is independent of the value of the\theta



Sample Questions

Question 1: Prove that sin2θ×(1+cot2θ)=1

Solution:

Taking the left hand side

=> We know that, \cosec^2 \theta=1+cot^2 \theta

=> \sin^2 \theta * (\cosec^2 \theta)

=> Also, \cosec \theta=\frac{1}{\sin \theta}        

=> \sin^2 \theta * (\frac{1}{\sin^2 \theta})      =1=R.H.S

Hence proved.

Question 2: Prove that  \mathbf {tan^2 \theta * (cosec^2 \theta-1)=1}

Solution:



Taking the left hand side

=> We know that, \cosec^2 \theta=1+\cot^2 \theta

=>\tan^2 \theta * (\cot^2 \theta)

=> Also, \cot \theta=\frac{1}{\tan \theta}

=> \tan^2 \theta * (\frac{1}{\tan^2 \theta})  =1=R.H.S

Hence proved.

Question 3: Find the value of \mathbf {sin^2 30\degree * (1+cot^2 30\degree)}

Solution:

=> sin^2 \theta * (1+cot^2 \theta)=1 , independent of the value of the value of \theta

Therefore, the value of sin^2 30\degree * (1+cot^2 30\degree)  is 1.



Question 4: Prove that \sec^2\theta*(1-\sin^2\theta)=1

Solution:

=> \sec^2\theta*(1-\sin^2\theta)

=> We know that, \sin^2 \theta+\cos^2\theta=1  , therefore, 1-\sin^2 \theta=\cos^2\theta

=> \sec^2\theta*\cos^2\theta

=> Also, \cos \theta=\frac{1}{\sec \theta}

=> \sec^2 \theta*\frac{1}{\sec^2 \theta}   =1




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