# PROVING TRIGONOMETRIC IDENTITIES PRACTICE PROBLEMS

Proving Trigonometric Identities Practice Problems :

Here we are going to see some practice problems on proving trigonometric identities.

## Proving Trigonometric Identities Practice Problems - Examples and solutions

Question 1 :

Determine whether each of the following is an identity or not.

(i) cos²θ + sec²θ = 2 + sin θ

Solution :

L.H.S

= cos²θ + sec²θ

= 1 - sin²θ + 1 + tan²θ

= 2 + tan²θ - sin²θ  ≠  R.H.S

Since L.H.S and R.H.S are not equal, the given statement is not an identity.

(ii)  cot²θ + cos θ = sin² θ

L.H.S

= cot²θ + cos θ

= cosec²θ - 1 + cos θ  ≠  R.H.S

Since L.H.S and R.H.S are not equal, the given statement is not an identity.

Question 2 :

Prove the following identities

(i)  sec²θ + cosec²θ = sec²θ  cosec²θ

Solution :

L.H.S

= sec²θ + cosec²θ  =  (1/cos²θ)+ (1/sin²θ)

taking L.C.M

=  (sin²θ + cos²θ)/(cos²θ sin²θ)

value of sin²θ + cos²θ = 1

=  1/(cos²θ sin²θ)

=  (1/cos²θ)(1/sin²θ)

=  sec²θ cosec²θ

(ii)  sin θ /(1 - cos θ) = cosec θ + cot θ

Solution :

L.H.S

= sin θ /(1 - cos θ)

Multiplying by the conjugate of denominator we get,

=  [sin θ /(1 - cos θ)] x [(1 + cos θ)/(1 + cos θ)]

instead of (1 + cos θ)/(1 + cos θ) we can write 1 - cos²θ by using the algebraic formula =  [sin θ (1 + cosθ)]/(1-cos²θ)

=  [sin θ (1 + cosθ)]/sin²θ

=  (1 + cosθ)]/sinθ

=  (1/sinθ) + (cosθ/sinθ)

=  cosec θ + cot θ

(iii)  √ (1-sin θ)/(1+sin θ) = sec θ - tan θ

Solution :

L.H.S

=  √(1-sin θ)/(1+sin θ)

=  (1-sin θ)/(1+sin θ) x (1-sin θ)/(1-sin θ)

=  (1-sin θ)²/[(1+sin θ) x (1-sin θ)]

=  (1-sin θ)²/(1²- sin²θ)

=  (1-sin θ)²/(cos²θ)

=  [(1-sin θ)/(cosθ)]²

=  [(1-sin θ)/(cosθ)]

=  [(1/cosθ)-(sin θ/cosθ)]

=  sec θ - tan θ

=  R.H.S After having gone through the stuff given above, we hope that the students would have understood, proving trigonometric identities practice problems.

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