Tan 45=1 because Tan45 can be written as sin45/cos45 =1/√2/1/√2==√2/√2=1

Explanation:

1. Trigonometric Functions:
Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of the sides of the triangle. The three main trigonometric functions are sine (sin), cosine (cos), and tangent (tan).

2. Tan Function:
The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In other words, tan(theta) = opposite/adjacent.

3. Tan 45:
When we evaluate the tangent of the angle 45 degrees (tan 45), we need to determine the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle with a 45-degree angle.

4. Right Triangle:
A right triangle is a triangle that has one angle measuring 90 degrees. In a right triangle, the side opposite the 90-degree angle is called the hypotenuse, and the other two sides are called the legs.

5. Special Right Triangle:
The 45-45-90 triangle is a special right triangle where the two legs are congruent and the angles are 45 degrees, 45 degrees, and 90 degrees. In this triangle, the ratio of the lengths of the sides is 1:1:√2.

6. Sin and Cos Functions:
In a 45-45-90 triangle, the length of the leg opposite the 45-degree angle is equal to the length of the leg adjacent to the 45-degree angle. Therefore, sin 45 degrees = 1/√2 and cos 45 degrees = 1/√2.

7. Tan 45:
To find tan 45 degrees, we divide the length of the side opposite the angle (1) by the length of the side adjacent to the angle (1). Therefore, tan 45 degrees = 1/1 = 1.

8. Simplification:
To simplify the expression, we can rewrite 1/√2 as (√2/√2) since dividing by a square root is the same as multiplying by its reciprocal. This simplifies to (√2/√2) = 1.

9. Conclusion:
In conclusion, tan 45 degrees is equal to 1 because the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a 45-45-90 triangle is 1:1. By simplifying the expression, we can also express tan 45 degrees as (√2/√2) = 1.

Toh pheer, why cos 45 and sin 45  =1/√2?