Question

If sinθ/secθ =1/2, find sinθ+secθ/sinθ-secθ

Answer 1

`\displaystyle (\sin \theta + \sec \theta)/(\sin \theta - \sec\theta) = -3`

Explanation:

We are given that:

`\displaystyle (\sin \theta)/(\sec \theta) = (1)/(2)`

And we want to find the value of:

`\displaystyle (\sin \theta + \sec \theta)/(\sin \theta - \sec\theta)`

From the second expression, we can divide both the numerator and denominator by sec(θ). Thus:

`\displaystyle = ( (\sin \theta + \sec \theta)/(\sec \theta) )/( (\sin \theta - \sec\theta)/(\sec \theta) )`

Simplify:

`\displaystyle = ((\sin \theta)/(\sec \theta) + 1)/((\sin\theta)/(\sec\theta) - 1)`

Since we know that sin(θ) / sec(θ) = 1 / 2:

`\displaystyle = (\left((1)/(2)\right)+1)/(\left((1)/(2)\right)-1)`

Evaluate:

`\displaystyle = ((3)/(2))/(-(1)/(2)) = -3`

Therefore:

`\displaystyle (\sin \theta + \sec \theta)/(\sin \theta - \sec\theta) = -3`