India's No. 1 – E-learning platform
Chapter 6 – Trigonometry – Text Book Solution
Practice Set 6.1| Q 6.1 | Page 131
sin²θ /Cos²θ + cosθ= secθ
Solution
We can start by manipulating the left-hand side of the equation using trigonometric identities.
sin²θ / (cos²θ + cosθ)
= sin²θ / cosθ(cosθ + 1)
= sinθ/cosθ * sinθ/cosθ / (1 + cosθ)
= tan²θ / (1 + cosθ)
Now, we can use the definition of secant:
secθ = 1/cosθ
Multiplying the numerator and denominator of the right-hand side by cos²θ, we get:
tan²θ / (1 + cosθ) * cos²θ/cos²θ
= sin²θ/cos²θ * cos²θ/(1 + cosθ)
= (sinθ/cosθ)² / (1/cosθ)(1 + cosθ)
= (tanθ)² / (1 + cosθ)
Now we can substitute this expression for the left-hand side of the original equation:
sin²θ / (cos²θ + cosθ) = (tanθ)² / (1 + cosθ)
= sec²θ – 1 / (1 + cosθ) (using the identity tan²θ + 1 = sec²θ)
= (1/cos²θ) – 1 / (1 + cosθ)
= (1 – cos²θ) / (cos²θ * (1 + cosθ))
= sin²θ / (cos²θ + cosθ)
Therefore, sin²θ / (cos²θ + cosθ) = secθ.
Practice set 6.1 |Q 6.1 | P 131
Click Here to see All the Textbook solution of Geometric Construction