PROVING TRIGONOMETRIC IDENTITIES QUESTIONS

Verify the identities :

Problem 1 :

(1 + sec² θ) / sec θ = 1 + cos²θ 

Solution :

= (1 + sec² θ) / sec θ

Using reciprocal identities,

sec² θ = 1/ cos² θ

= (1 + (1/ cos² θ)) / (1/cos θ)

= [ (cos² θ + 1) / cos θ ] / (1/cos θ)

= [ (cos² θ + 1) / cos θ ]  x  (cos θ / 1)

= 1 + cos² θ

Hence it is proved.

Problem 2 :

(sin θ / cos θ) + (cos θ / sin θ) = 1/sin θ cos θ

Solution :

= (sin θ / cos θ) + (cos θ / sin θ)

Since two fractions are added, we have to make the denominators same and add the numerators.

= (sin θ/cos θ) x (sin θ/sin θ) + (cos θ/sin θ) x (cos θ/cos θ)

= (sin² θ/sin θ cos θ) + (cos² θ/sin θcos θ)

= (sin² θ + cos² θ) / sin θ cos θ

= 1 / sin θ cos θ

Hence it is proved.

Problem 3 :

Solution :

Problem 4 :

sin²𝜃(1 + cot²𝜃) = 1

Solution :

sin²𝜃(1 + cot²𝜃) = 1

L.H.S

= sin²𝜃(1 + cot²𝜃)

= sin²𝜃(1 + (cos²𝜃/sin²𝜃) )

= sin²𝜃((sin²𝜃+cos²𝜃) / sin²𝜃 )

= (sin²𝜃+cos²𝜃)

Using Pythagorean identities, the value of (sin²𝜃+cos²𝜃) is 1

= 1

Hence proved.

Problem 5 :

Solution :