Question

Prove that `(sin(90^circ - θ))/(cos θ) + (cos(90^circ - θ))/(sin θ) = 2`.

Answer 1

Given: Let θ be an angle. Consider `sin(90^circ - θ)/(cos θ) + (cos (90^circ - θ))/(sin θ)`

To Prove: `sin(90^circ - θ)/(cos θ) + (cos (90^circ - θ))/(sin θ) = 2`

Proof [Step-wise]:

1. Use the complementary-angle identities:

sin (90° – θ) = cos θ and cos (90° – θ) = sin θ

2. Substitute these into the expression:

`sin(90^circ - θ)/(cos θ)`

= `(cos θ)/(cos θ)`

= 1

3. Similarly, `cos(90^circ - θ)/(sin θ)`

= `(sin θ)/(sin θ)`

= 1

4. Add the two results:

1 + 1 = 2

Hence, `sin(90^circ - θ)/(cos θ) + (cos (90^circ - θ))/(sin θ) = 2`, as required.