If tan theta + cot theta = 7 then sin theta + cos theta = ?

√ (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