Question

Prove that 1+sinθ−cosθ/1+sinθ+cosθ=tanθ/2

Answer 1

To prove the equation 1 + sinθ - cosθ / 1 + sinθ + cosθ = tanθ / 2, we can simplify the left side using trigonometric identities and show that it is equal to the right side.

Step-by-step explanation:

To prove the equation 1 + sinθ - cosθ / 1 + sinθ + cosθ = tanθ / 2, we can start by simplifying the left side of the equation using trigonometric identities:

1. Add the fractions together: (1 + sinθ - cosθ) / (1 + sinθ + cosθ)
2. Use the identity cos^2θ + sin^2θ = 1 to simplify the numerator: (1 + 2sinθ - 2sinθcosθ) / (1 + sinθ + cosθ)
3. Divide both the numerator and denominator by cosθ: (1/cosθ + 2sinθ/cosθ - 2sinθcosθ/cosθ) / (1/cosθ + sinθ/cosθ + cosθ/cosθ)
4. Simplify further: (secθ + 2tanθ - 2sinθ) / (secθ + tanθ + 1)
5. Use the identity tanθ = sinθ/cosθ to rewrite the equation: (secθ + 2sinθ/cosθ - 2sinθ) / (secθ + sinθ/cosθ + 1)
6. Simplify again: (secθ + 2sinθ - 2sinθ) / (secθ + sinθ + cosθ)
7. Cancel out the terms: secθ / (secθ + sinθ + cosθ)
8. Use the identity secθ = 1/cosθ: 1 / (1 + sinθ + cosθ)

Therefore, the left side of the equation simplifies to 1 / (1 + sinθ + cosθ), which is equal to the right side of the equation, tanθ / 2.