sin θ/ (1 + cos θ) + (1 + cos θ)/ sin θ = 2 cosec θ. Prove the following statement

Solution:

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

By taking LCM we get

= [sin² θ + (1 + cos θ)² ]/[sin θ (1 + cos θ)]

Using the algebraic identity

(a + b)² = a² + b² + 2ab

= [sin² θ + 1 + cos² θ + 2 cos θ]/ [sin θ (1 + cos θ)]

As sin² θ + cos² θ = 1

= [1 + 1 + 2 cos θ]/ [sin θ (1 + cos θ)]

= [2 + 2 cos θ]/ [sin θ (1 + cos θ)]

Taking out the common terms in numerator

= 2[1 + cos θ]/ [sin θ (1 + cos θ)]

So we get

= 2/ sin θ

As 1/sin θ = cosec θ

= 2 cosec θ

= RHS

Therefore, it is proved.

✦ Try This: Prove the following:

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

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8

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NCERT Exemplar Class 10 Maths Exercise 8.3 Problem 1

sin θ/ (1 + cos θ) + (1 + cos θ)/ sin θ = 2 cosec θ. Prove the following statement

Summary:

The sine function is written as the ratio of the length of the perpendicular and hypotenuse of the right-angled triangle. It is proved that sin θ/ (1 + cos θ) + (1 + cos θ)/ sin θ = 2 cosec θ

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☛ Related Questions:

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