Answer:
√(9/7)
Step-by-step explanation:
tan θ + cot θ = 7
Write tan and cot in terms of sin and cos.
(sin θ / cos θ) + (cos θ / sin θ) = 7
Multiply both sides by sin θ cos θ
sin² θ + cos² θ = 7 sin θ cos θ
Use Pythagorean identity.
1 = 7 sin θ cos θ
1/7 = sin θ cos θ
Remember that, we'll use it later. Go back to this step:
sin² θ + cos² θ = 7 sin θ cos θ
This time, add 2 sin θ cos θ to both sides.
sin² θ + 2 sin θ cos θ + cos² θ = 9 sin θ cos θ
Factor.
(sin θ + cos θ)² = 9 sin θ cos θ
Substitute.
(sin θ + cos θ)² = 9 / 7
Take square root.
sin θ + cos θ = √(9/7)
Option: rationalize the denominator.
sin θ + cos θ = 3√7 / 7
