If tanθ = 1 then, find the values of (sinθ + cosθ)/(secθ+cosecθ).

Solution

Given tan(θ) = 1, we can draw a right triangle with one acute angle θ and label the opposite and adjacent sides as equal, so that opposite = adjacent = 1. Then, using the Pythagorean theorem, we can find the hypotenuse:

1² + 1² = c²

c = √2

Now we can use the definitions of the trigonometric functions to find the values of sin(θ), cos(θ), sec(θ), and cosec(θ):

sin(θ) = opposite/hypotenuse = 1/√2 = √2/2

cos(θ) = adjacent/hypotenuse = 1/√2 = √2/2

sec(θ) = hypotenuse/adjacent = √2/1 = √2

cosec(θ) = hypotenuse/opposite = √2/1 = √2

Using these values, we can evaluate the expression (sin(θ) + cos(θ))/(sec(θ) + cosec(θ)):

(sin(θ) + cos(θ))/(sec(θ) + cosec(θ)) = [(√2/2) + (√2/2)]/[(√2) + (√2)]

= (√2)/(2√2)

= 1/2

Therefore, (sin(θ) + cos(θ))/(sec(θ) + cosec(θ)) = 1/2 when tan(θ) = 1.