Simplify cos(y)csc(pi/2-y)
The question requires you to simplify the mathematical expression given, which consists of trigonometric functions cosine (cos) and cosecant (csc). The expression specifically is cos(y)csc(pi/2-y), and to simplify it, you will need to recognize and apply trigonometric identities and properties, such as the complementary angle identity csc(90° - θ) = sec(θ), where θ is an angle measured in degrees or, equivalently, csc(π/2 - y) = sec(y) when the angle is measured in radians.
$cos \left(\right. y \left.\right) csc \left(\right. \frac{\pi}{2} - y \left.\right)$
Answer
Solution:
Step 1:
Express $csc\left(\frac{\pi}{2} - y\right)$ using sine and cosine functions: $cos(y) \cdot \frac{1}{sin(\frac{\pi}{2} - y)}$.
Step 2:
Merge $cos(y)$ and $\frac{1}{sin(\frac{\pi}{2} - y)}$: $\frac{cos(y)}{sin(\frac{\pi}{2} - y)}$.
Step 3:
Transform the expression $\frac{cos(y)}{sin(\frac{\pi}{2} - y)}$ into a multiplication format: $cos(y) \cdot \frac{1}{sin(\frac{\pi}{2} - y)}$.
Step 4:
Represent $cos(y)$ as a fraction with a denominator of $1$: $\frac{cos(y)}{1} \cdot \frac{1}{sin(\frac{\pi}{2} - y)}$.
Step 5:
Proceed to simplify the expression.
Step 5.1:
Divide $cos(y)$ by $1$: $cos(y) \cdot \frac{1}{sin(\frac{\pi}{2} - y)}$.
Step 5.2:
Change $\frac{1}{sin(\frac{\pi}{2} - y)}$ back to $csc(\frac{\pi}{2} - y)$: $cos(y) \cdot csc(\frac{\pi}{2} - y)$.
Knowledge Notes:
The problem involves simplifying a trigonometric expression that contains both cosine and cosecant functions. Here are the relevant knowledge points:
Trigonometric Functions: The basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). These functions are fundamental in the study of triangles, waves, and oscillations.
Cosecant Function: The cosecant (csc) is the reciprocal of the sine function. That is, $csc(\theta) = \frac{1}{sin(\theta)}$.
Complementary Angles: In trigonometry, two angles are complementary if they add up to $\frac{\pi}{2}$ radians (or 90 degrees). The sine of an angle is equal to the cosine of its complement, and vice versa: $sin(\frac{\pi}{2} - \theta) = cos(\theta)$ and $cos(\frac{\pi}{2} - \theta) = sin(\theta)$.
Simplification: Simplifying an expression involves rewriting it in a more concise or more easily understandable form, often by combining like terms or using identities to replace parts of the expression with simpler equivalents.
Trigonometric Identities: These are equations involving trigonometric functions that are true for all values of the variables where both sides of the equation are defined. They are useful in simplifying expressions and solving equations involving trigonometric functions.
In this problem, the process involves using the complementary angle identity and the definition of the cosecant function to simplify the given expression. The final result is a simplified version of the original expression, which is easier to interpret or use in further calculations.
