Here we will prove the problems on trigonometric identities. In an identity there are two sides of the equation, one side is known as ‘left hand side’ and the other side is known as ‘right hand side’ and to prove the identity we need to use logical steps showing that one side of the equation ends up with the other side of the equation.

Proving the problems on trigonometric identities:

L.H.S = (1 - sin A)/(1 + sin A)

= (1 - sin A)

= (1 - sin A)

= (1 - sin A)

= {(1 - sin A)/cos A}

= (1/cos A - sin A/cos A)

= (sec A – tan A)

L.H.S.= √{(sec θ – 1)/(sec θ + 1)}

= √[{(sec θ - 1) (sec θ - 1)}/{(sec θ + 1) (sec θ - 1)}]; [multiplying numerator and denominator by (sec θ - l) under radical sign]

= √{(sec θ - 1)

=√{(sec θ -1)

= (sec θ – 1)/tan θ

= (sec θ/tan θ) – (1/tan θ)

= {(1/cos θ)/(sin θ/cos θ)} - cot θ

= {(1/cos θ) × (cos θ/sin θ)} - cot θ

= (1/sin θ) - cot θ

= cosec θ - cot θ = R.H.S.

L.H.S = tan

= tan

= (sec

= (sec

= sec

More problems on trigonometric identities are shown where one side of the identity ends up with the other side.

L.H.S = cos θ/(1 - tan θ) + sin θ/(1 - cot θ)

= cos θ/{1 - (sin θ/cos θ)} + sin θ/{1 - (cos θ/sin θ)}

= cos θ/{(cos θ - sin θ)/cos θ} + sin θ/{(sin θ - cos θ/sin θ)}

= cos

= (cos

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

= (cos θ + sin θ) = R.H.S.

We have,

1/(csc A - cot A) + 1/(csc A + cot A)

= (csc A + cot A + csc A - cot A)/(csc

= (2 csc A)/1; [since, csc

= 2/sin A; [since, csc A = 1/sin A]

Therefore,

1/(csc A - cot A) + 1/(csc A + cot A) = 2/sin A

⇒ 1/(csc A - cot A) + 1/(csc A + cot A) = 1/sin A + 1/sin A

Therefore, 1/(csc A - cot A) - 1/sin A = 1/sin A - 1/(csc A + cot A)

L.H.S = (tan θ + sec θ - 1)/(tan θ - sec θ + 1)

= [(tan θ + sec θ) - (sec

= {(tan θ + sec θ) - (sec θ + tan θ) (sec θ - tan θ)}/(tan θ - sec θ + 1)

= {(tan θ + sec θ) (1 - sec θ + tan θ)}/(tan θ - sec θ + 1)

= {(tan θ + sec θ) (tan θ - sec θ + 1)}/(tan θ - sec θ + 1)

= tan θ + sec θ

= (sin θ/cos θ) + (1/cos θ)

= (sin θ + 1)/cos θ

= (1 + sin θ)/cos θ = R.H.S.

**●** **Trigonometric Functions**

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