Cos theata ( sin theata 90- theata ) cos 2teata
cos theata = sin theata = 1
cos 2 ,sin ×2 theta=1

Introduction

In trigonometry, cosine (cos) is a mathematical function that relates the angle of a right triangle to the ratios of its sides. The equation cos^2(theta) = 1 involves squaring the cosine of an angle and setting it equal to 1. To understand why this equation holds true, we will explore the properties of cosine and its relationship with other trigonometric functions.

The Basics of Cosine

The cosine of an angle theta (cos(theta)) can be defined as the ratio of the length of the adjacent side of a right triangle to the hypotenuse. It is calculated by dividing the length of the adjacent side by the length of the hypotenuse.

The Pythagorean Identity

One of the fundamental trigonometric identities is the Pythagorean identity, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it can be expressed as:

c^2 = a^2 + b^2

where c represents the hypotenuse, and a and b represent the other two sides of the triangle. This identity holds true for all right triangles.

The Relationship between Cosine and Pythagorean Identity

In a right triangle, the cosine of an angle theta is defined as the ratio of the adjacent side to the hypotenuse. Let's denote the adjacent side as a and the hypotenuse as c. Using the Pythagorean identity, we can rewrite it as:

c^2 = a^2 + b^2

Squaring both sides of this equation, we get:

(c^2)^2 = (a^2 + b^2)^2

Expanding the right-hand side of the equation, we have:

c^4 = a^4 + 2a^2b^2 + b^4

Substituting the Relationship

Now, let's substitute the values of a^2 and b^2 with their trigonometric equivalents. Since cos^2(theta) = a^2 and sin^2(theta) = b^2, we can rewrite the equation as:

c^4 = cos^4(theta) + 2(cos^2(theta))(sin^2(theta)) + sin^4(theta)

Using the trigonometric identity sin^2(theta) = 1 - cos^2(theta), we further simplify the equation as:

c^4 = cos^4(theta) + 2(cos^2(theta))(1 - cos^2(theta)) + (1 - cos^2(theta))^2

Expanding and combining like terms, we have:

c^4 = cos^4(theta) + 2cos^2(theta) - 2cos^4(theta) + 1 - 2cos^2(theta) + cos^4(theta)

Simplifying the equation, we get:

c^4 = 1

Taking the square root of both sides, we obtain:

c^2 = 1

Conclusion

From this derivation, we can see that the equation cos^2(theta) = 1 holds true based on the properties of cosine, the Pythagorean identity, and the relationship between cosine and sine. It is an essential identity in trigonometry and has various applications in solving