Trig Identities Problems

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Trigonometric identities are the statements or the equations containing the trigonometric functions, and these functions are put together using different operations. Unlike equations, these trigonometric identities are not solved, but these identities are proved. In order to prove the identities, we have to get to a conclusion that the expression in the left-hand side of the equation is the same as the expression in the right-hand side of the equation. By proving both sides of the equation to be the same, we can conclude that the trigonometric identity is proved!

Example 1: Prove the given trigonometric identity: cot(θ)/cosec(θ) = cos(θ).

In order to prove the above given trigonometric identity, we have to first start by picking any side of the equation.

Here let’s start with the left-hand side of the equation->cot(θ)/cosec(θ)

We can also write the above expression as: cot(θ)/cosec(θ) = [cos(θ)/sin(θ)]/ [1/sin(θ)]

Now taking the reciprocal we get->[cos(θ)/sin(θ)] * [sin(θ)/1]

Now, sin(θ) gets cancelled, and we get ->cos(θ) = right-hand side of the equation!

Hence proved!

Example 2: Prove the given trigonometric identity: sec(θ) * cot(θ) = cosec(θ).

In order to prove the above given trigonometric identity, we have to first start by picking any side of the equation.

Here let’s start with the left-hand side of the equation ->sec(θ) * cot(θ)

We can also write the above expression as: sec(θ) * cot(θ) = [1/cos(θ)] *  [cos(θ)/sin(θ)]

Here the cos(θ) present in the numerator and the denominator gets cancelled and we get ->[1/sin(θ)] .

This implies: 1/sin(θ) = cosec(θ) = right-hand side of the equation!

Hence proved!
 


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