Question

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

`cot θ tan (90^circ - θ) - sec (90^circ - θ)"cosec"  θ + sin^2 65^circ + sin^2 25^circ + sqrt(3) tan 5^circ tan 45^circ tan 85^circ`

Answer 1

Given: `cot θ tan (90^circ - θ) - sec (90^circ - θ)"cosec"  θ + sin^2 65^circ + sin^2 25^circ + sqrt(3) tan 5^circ tan 45^circ tan 85^circ`

Step-wise calculation:

1. tan(90° – θ) = cot θ

So cot θ · tan(90° – θ) = cot θ · cot θ = cot²θ

2. sec(90° – θ) = cosec θ

So sec(90° – θ)·cosec θ = cosec θ·cosec θ = cosec²θ

3. Therefore the first two terms give cot²θ – cosec²θ. 

Using cosec²θ = 1 + cot²θ, we get 

cot²θ – cosec²θ 

= cot²θ – (1 + cot²θ) 

= –1

4. sin²65° + sin²25°:

Since 65° + 25° = 90° 

sin²α + sin²(90° – α) = 1 

So sin²65° + sin²25° = 1.

5. `sqrt(3)·tan 5^circ·tan 45^circ·tan 85^circ`: 

tan 45° = 1 and tan 85° = tan(90° – 5°) = cot 5°

So tan 5°·tan 85° = tan 5°·cot 5° = 1. 

Hence this term = `sqrt(3)·1·1 = sqrt(3)`.

6. Combine results:

(first two terms) –1 + (sin² sum) 1 + (last term) `sqrt(3)`

= `-1 + 1 + sqrt(3)` 

= `sqrt(3)`

The value of the expression is `sqrt(3)`.