If secθ = √2, then what is the value of cosecθ?

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Given secθ = √2.

Recall that secθ = 1/cosθ, so cosθ = 1/secθ = 1/√2 = √2/2.

Using the Pythagorean identity sin²θ + cos²θ = 1, we find sinθ:

sin²θ = 1 - cos²θ = 1 - (√2/2)² = 1 - 1/2 = 1/2.

Therefore, sinθ = ±√(1/2) = ±√2/2.

Cosecθ = 1/sinθ = ±1/(√2/2) = ±√2.

Since secθ is positive (√2), θ lies in the first or fourth quadrant. In the first quadrant, sinθ is positive, so cosecθ = √2.

Hence, cosecθ = √2.

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