Question

Without using trigonometric tables, find the value of the expression:

`(cos(90^circ - θ) sec(90^circ - θ) tan θ)/("cosec"(90^circ - θ)sin(90^circ - θ)cot(90^circ - θ)) + (tan(90^circ - θ))/(cot θ)`

Answer 1

Given: `(cos(90^circ - θ) sec(90^circ - θ) tan θ)/("cosec"(90^circ - θ)sin(90^circ - θ)cot(90^circ - θ)) + (tan(90^circ - θ))/(cot θ)`

Step-wise calculation:

1. Use co-function identities:

sin(90° – θ) = cos θ

cos(90° – θ) = sin θ

tan(90° – θ) = cot θ

sec(90° – θ) = cosec θ

And cosec(90° – θ) = sec θ

2. Substitute into the numerator:

cos(90° – θ)·sec(90° – θ)·tan θ 

= `(sin θ)·(1/sin θ)·tan θ`

= tan θ

3. Substitute into the denominator:

cosec(90° – θ)·sin(90° – θ)·cot(90° – θ) 

= (sec θ)·(cos θ)·(tan θ)

= tan θ

So the first fraction = `(tan θ)/(tan θ)` = 1.

4. Second term: `(tan(90^circ - θ))/(cot θ)`

= `(cot θ)/(cot θ)` 

= 1

5. Sum = 1 + 1 = 2.

The value of the expression is 2, for values of θ where the expression is defined (i.e., sin θ ≠ 0 and cos θ ≠ 0, so θ is not an integer multiple of 90°).