Prove the identity: (sin θ)/(1 + cos θ) + (1 + cos θ)/(sin θ) = 2 cosec θ.

Step 1: Combine fractions on the Left Hand Side

To add the two fractions on the Left Hand Side (LHS), we find a common denominator, which is the product of the individual denominators: $\sin \theta (1 + \cos \theta)$. We then cross-multiply the numerators and add them.

Step 2: Expand the numerator

Next, we expand the term $(1 + \cos \theta)^2$ in the numerator using the algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$. Here, $a=1$ and $b=\cos \theta$.

Step 3: Apply Pythagorean Identity

We know the fundamental trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$. We substitute this into the numerator, which simplifies the expression.

Step 4: Factor and simplify

We factor out 2 from the numerator. Since $(1 + \cos \theta)$ appears in both the numerator and the denominator, we can cancel this common term, simplifying the fraction.

Step 5: Express in terms of cosecant

Finally, we use the reciprocal identity $\operatorname{cosec} \theta = \frac{1}{\sin \theta}$ to express the simplified LHS in terms of cosecant, which matches the Right Hand Side (RHS) of the identity.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

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