Question

Trigeonometric Identities

Exercise 13 A

Question 30

(cos ∅ × cosec theta - sin ∅ × sec theta) / (cos ∅ + sin ∅) = cosec ∅ - sec ∅​

Answer 1

To prove:-

(cosθcosecθ - sinθsecθ)/(cosθ + sin θ)

= cosecθ - secθ

Solution:-

We know that :-

• cosec θ = 1/sinθ

• sec θ = 1/cosθ

________________________________

LHS :-

= [cosθcosecθ - sinθsecθ]/[cosθ + sinθ]

= [cosθ.1/sinθ - sinθ.1/cosθ]/[cosθ + sinθ]

= [cosθ/sinθ - sinθ/cosθ]/[cosθ + sinθ]

= [cos²θ - sin²θ]/[sinθcosθ(cosθ + sinθ)]

= [(cosθ - sinθ)]/[sinθcosθ]

= [cosθ/sinθcosθ] - [sinθ/sinθcosθ]

= [1/sinθ] - [1/cosθ]

= cosecθ - secθ

RHS :-

= cosecθ - secθ

∴ LHS = RHS proved!

Some Extra Information:-

Some basic identities are :-

→ sin²θ + cos²θ = 1

→ 1 + cot²θ = cosec²θ

→ tanθ × cotθ = 1

→ 1 + tan²θ = sec²θ

→ tanθ = sinθ/cosθ

→ cotθ = cosθ/sinθ

Answer 2

To prove:-

(cosθcosecθ - sinθsecθ)/(cosθ + sin θ)

= cosecθ - secθ

Solution:-

We know that :-

• cosec θ = 1/sinθ

• sec θ = 1/cosθ

________________________________

LHS :-

= [cosθcosecθ - sinθsecθ]/[cosθ + sinθ]

= [cosθ.1/sinθ - sinθ.1/cosθ]/[cosθ + sinθ]

= [cosθ/sinθ - sinθ/cosθ]/[cosθ + sinθ]

= [cos²θ - sin²θ]/[sinθcosθ(cosθ + sinθ)]

= [(cosθ - sinθ)]/[sinθcosθ]

= [cosθ/sinθcosθ] - [sinθ/sinθcosθ]

= [1/sinθ] - [1/cosθ]

= cosecθ - secθ

RHS :-

= cosecθ - secθ

∴ LHS = RHS proved!

Some Extra Information:-

Some basic identities are :-

→ sin²θ + cos²θ = 1

→ 1 + cot²θ = cosec²θ

→ tanθ × cotθ = 1

→ 1 + tan²θ = sec²θ

→ tanθ = sinθ/cosθ

→ cotθ = cosθ/sinθ