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Chapter 6 – Trigonometry – Text Book Solution
Practice Set 6.1| Q 6.2 | Page 131
Prove that Cos²θ (1+tan²θ )=1
Solution
We can start with the identity:
1 + tan²θ = sec²θ
Dividing both sides by sec²θ, we get:
1/sec²θ + tan²θ/sec²θ = 1
Using the identity sec²θ = 1/cos²θ, we can rewrite this as:
cos²θ/(cos²θ)² + sin²θ/(cos²θ)² = 1
Simplifying the left-hand side using the identity sin²θ + cos²θ = 1, we get:
1/(cos²θ)² = 1
Multiplying both sides by (cos²θ)², we get:
1 = (cos²θ)²
Expanding the left-hand side, we get:
cos²θ * (1 + tan²θ) = 1
Multiplying both sides by cos²θ, we get:
cos⁴θ * (1 + tan²θ) = cos²θ
Dividing both sides by cos⁴θ, we get:
1 + tan²θ = 1/cos²θ
cos²θ + cos²θ tan²θ = 1
Subtracting cos²θ tan²θ from both sides, we get:
cos²θ (1 + tan²θ) = 1
Therefore, we have shown that Cos²θ (1 + tan²θ) = 1.
Practice set 6.1 |Q 6.2 | P 131
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