Question

if tan theta + cot theta equal to 2 where theta is less than 90 degree and greater than zero degree then find the value of sin 15 theta + cos 45 theta

Answer 1

Given conditions ⇒

tan θ + Cot θ = 2
We know,
Cot θ = 1/tan θ
∴ tan θ + 1/tan θ = 2
∴ (tan² θ + 1)/tan θ = 2
∴ tan²θ + 1 = 2 tanθ 
∴ tan²θ - 2 tanθ + 1 = 0
⇒ (tan θ  - 1)² = 0
⇒ tanθ - 1 = √0
⇒ tanθ - 1 = 0
∴ tanθ = 1
∴ tanθ =1
∴ tanθ = tan 45°
On Comparing,
 θ = 45°

A/c to the Question, 
 90° < θ > 0°
∴  45° lies between 90° and 0°, Hence the Value of the θ is correct.

Now,
Sin 15θ + Cos 45θ = Sin(15 × 45) + Cos (45 × 45)
= Sin 675 + Cos 2025
675 can be written as (630 + 45) and 2025 can be written as (1980 + 45).
= Sin(630 + 45) + Cos (1980 + 45)
= Sin 630 Cos 45 + Cos 630 Sin 45 + Cos 1980 Cos 45 - Sin 1980 Sin 45
We know, 
Sin 45 = Cos 45 = 1/√2, Sin 630 = -1, Cos 630 = 0,Cos 1980 = -1, & Sin 1980 = 0
On substituting these values in the Equation, wee will get,

= (-1 × 1/√2) + (0 × 1/√2) + (-1 × 1/√2) - (0 × 1/√2)
= -1/√2 + 0  -1/√2 - 0
= -1/√2 - 1/√2
= (-1 -1) ÷ √2
= -2/√2
= -√2


Hence, the value of the Given Trignometric Equation will be -√2.


Hope it helps.

Answer 2

If , then the value of is