# Prove tan^{2} θ – (1/cos^{2} θ) + 1 = 0

It is basically a study of the properties of the triangle and trigonometric function and its application in various cases. It helps find the angles and missing sides of the triangle with the help of trigonometric ratios. Generally, some angle’s values are known from which we get the value of other angles. The commonly known angle angles are 0°, 30°, 45°, 60°, and 90°.

Trigonometry is known for its identities. The trigonometric identities are commonly used for rewriting trigonometrical expressions to simplify an expression, to get a more useful sort of an expression, or to unravel an equation. Problems in which a one-dimensional plane is used are done with the help of plane trigonometry. Applications to similar problems in additional than one plane of three-dimensional space are considered in trigonometry.

### Trigonometric Ratios

Trigonometric ratios are the ratios between sides of a right-angled triangle. These ratios are given by the following trigonometric functions of the known angle, where perpendicular, base, and hypotenuse refer to the lengths of the sides in the below figure,

The hypotenuse is the side opposite to the right angle, it is the largest side in the triangle. The base is the side that contains the angle. Perpendicular is the side opposite to the given angle.

The six important trigonometric functions (trigonometric ratios) are calculated using the below formulas and considering the above figure. It is necessary to get knowledge about the sides of the right triangle because it defines the set of important trigonometric functions.

- The sine of θ is written as sinθ and defined as the ratio
**sinθ = perpendicular/hypotenuse** - The cosine of θ is written as cosθ and defined as the ratio
**cosθ = base/hypotenuse** - The tangent of θ is written as tanθ and defined as the ratio
**tanθ = perpendicular/base = sinθ/cosθ**

NoteThe reciprocals of sine, cosine, and tangents also have names: they are cosecant, secant, and cotangent.

- The cosecant of θ is written as cosecθ and defined as
**cosecθ = 1/sinθ** - The secant of θ is written as secθ and defined as
**secθ = 1/cosθ** - The cotangent of θ is written as cotθ and defined as
**cotθ = 1/tanθ**

**Trigonometric ratio table**

Angles | 0° | 30° | 45° | 60° | 90° |

Sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |

Cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |

Tan θ | 0 | 1/√3 | 1 | √3 | ∞ |

Cosec θ | ∞ | 2 | √2 | 2/√3 | 1 |

Sec θ | 1 | 2/√3 | √2 | 2 | ∞ |

Cot θ | ∞ | √3 | 1 | 1/√3 | 0 |

There are three Pythagorean Identities,

**sin**^{2}θ + cos^{2}θ = 1**tan**^{2}θ + 1 = sec^{2}θ**cot**^{2}θ + 1 = cosec^{2}θ

### Prove: tan^{2} θ – (1/cos^{2} θ) + 1 = 0

**Solution:**

To prove – tan

^{2}θ – (1/cos^{2}θ) + 1 = 0It is known,

⇒ 1/cosθ = secθ

So the equation becomes,

⇒ tan

^{2}θ – sec^{2}θ + 1 = 0From the Pythagorean identity, tan

^{2}θ + 1 = sec^{2}θSo the equation becomes,

⇒ sec

^{2}θ – sec^{2}θ which is clearly equal to 0⇒ tan

^{2}θ – (1/cos^{2}θ) + 1 = 0Hence Proved

### Sample Problems

**Question 1: Find the value of sin ^{2}x – cos^{2}x in terms of sinx**

**Solution:**

⇒ sin

^{2}x + cos^{2}x = 1 [From pythagorean identity]So,

cos

^{2}x = 1 – sin^{2}xPutting this,

sin

^{2}x – (1 – sin^{2}x) which is equal to2sin

^{2}x – 1

**Question 2: Find the value of 12tan ^{2}x – 12sec^{2}x + 12**

**Solution:**

⇒ 12tan

^{2}x – 12sec^{2}x + 12 ⇢ 12(tan^{2}x – sec^{2}x) + 1It is known, sec

^{2}x – tan^{2}x = 1 [from pythagorean identity]So, the equation becomes 12(-1) + 12 = 0

12tan

^{2}x – 12sec^{2}x + 12 = 0

**Question 3: If tanx = 3 find the value of sec ^{2}x + cosec^{2}x**

**Solution:**

⇒ It is known, sec

^{2}x = 1 + tan^{2}x [from pythagorean identity]cosec

^{2}x = 1 + cot^{2}x [from pythagorean identity]Also, cotx = 1/tanx

Equation becomes 2 + tan

^{2}x + cot^{2}xPutting the values,

2 + 9 + 1/9 = 100/9

sec

^{2}x + cosec^{2}x, when tanx = 3 ⇒ 100/9