Question

Solve for x (x lies between 0° and 90°

tan²x + 3 = 3 sec x

Answer 1

Solution:

To Determine:- The values of x.

Given Equation :-

→ tan²(x) + 3 = 3 sec(x) [0° ≤ x ≤ 90°]

We know that :-

→ sec²(x) - tan²(x) = 1

→ tan²(x) = sec²(x) - 1

Now, the equation becomes :-

→ sec²(x) - 1 + 3 = 3 sec(x)

→ sec²(x) - 3 sec(x) + 2 = 0

By splitting the middle term, we get :-

→ sec²(x) - sec(x) - 2 sec(x) + 2 = 0

→ sec(x)[sec(x) - 1] - 2[sec(x) - 1] = 0

→ [sec(x) - 2] × [sec(x) - 1] = 0

→ sec(x) = 1, 2

But sec 0° = 1 and sec 60° = 2

Therefore :-

→ x = 0°, 60° (Answer)

Learn More:

1. Relationship between sides and T-Ratios.

• sin θ = Height/Hypotenuse
• cos θ = Base/Hypotenuse
• tan θ = Height/Base
• cot θ = Base/Height
• sec θ = Hypotenuse/Base
• cosec θ = Hypotenuse/Height

2. Square formulae.

• sin²θ + cos²θ = 1
• cosec²θ - cot²θ = 1
• sec²θ - tan²θ = 1

3. Reciprocal Relationship.

• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

4. Cofunction identities.

• sin(90° - θ) = cos θ
• cos(90° - θ) = sin θ
• cosec(90° - θ) = sec θ
• sec(90° - θ) = cosec θ
• tan(90° - θ) = cot θ
• cot(90° - θ) = tan θ

5. Even odd identities.

• sin -θ = -sin θ
• cos -θ = cos θ
• tan -θ = -tan θ