Question
If sec θ + tan θ = 5, then find the value of sin
θ.Solution
Given sec θ + tan θ = 5. Let sec θ = x and tan θ = y. We have x + y = 5 and also the identity sec² θ - tan² θ = 1. This gives x² - y² = 1. Now, solving these two equations: (x - y)(x + y) = 1, so (5)(x - y) = 1, giving x - y = 1/5. Now, 2x = 5 + 1/5 = 26/5, so x = 13/5. Thus, sec θ = 13/5, so cos θ = 5/13. Now, sin θ = √(1 - cos² θ) = √(1 - (5/13)²) = √(144/169) = 12/13. Answer: b) 12/13.
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