**Verification of the Identity:**\[\frac{\cos(u) \sec(u)}{\tan(u)} = \cot(u)\]**Objective:** Use a Reciprocal Identity to rewrite the expression and then simplify.Step-by-Step Solution:1. Begin with the given expression:\[\frac{\cos(u) \sec(u)}{\tan(u)}\]2. Apply the Reciprocal Identity. Expand \(\tan(u)\) as \(\frac{\sin(u)}{\cos(u)}\):\[\frac{\cos(u) \sec(u)}{\frac{\sin(u)}{\cos(u)}}\]Red note with an "X" indicates an incorrect approach by rewriting the expression as \((\cos(u) \sec(u))(\tan(u))\). This approach is not correct.3. Simplify the multiplication:\[= \cos(u) \cdot \sec(u) \cdot \frac{\cos(u)}{\sin(u)}\]4. Recognize that \(\sec(u) = \frac{1}{\cos(u)}\):\[= \cos(u) \cdot \frac{1}{\cos(u)} \cdot \frac{\cos(u)}{\sin(u)}\]5. Simplify the intermediate expression:\[= \cos(u) \cdot \frac{1}{\cos(u)} \cdot \frac{\cos(u)}{\sin(u)}\]So, \[= \cos(u) \cdot \frac{\cos(u)}{\cos(u)} \cdot \frac{1}{\sin(u)}\]6. Simplify \(\frac{\cos(u)}{\cos(u)}\) to 1:\[= 1 \cdot \frac{1}{\sin(u)}\]7. Thus, the expression simplifies to the cotangent function:\[= \cot(u) \quad \checkmark\] # Verifying Trigonometric Identities## Verifying the Identity:Given the identity:\[ \frac{\sec(t) - \cos(t)}{\sec(t)} = \sin^2(t) \]We aim to verify this identity using fundamental trigonometric identities.## Step-by-Step Verification### Step 1: Use a Reciprocal IdentityFirstly, we use the reciprocal identity for \(\sec(t)\):\[ \sec(t) = \frac{1}{\cos(t)} \]Rewrite the given expression in terms of cosine only:\[ \frac{\sec(t) - \cos(t)}{\sec(t)} \]Substitute the reciprocal identity \(\sec(t) = \frac{1}{\cos(t)}\):\[ \frac{\frac{1}{\cos(t)} - \cos(t)}{\frac{1}{\cos(t)}} \]### Step 2: Simplify the ExpressionFactor out \(\frac{1}{\cos(t)}\):\[ \left( \frac{1}{\cos(t)} \right) \left( \frac{1}{\cos(t)} - \cos(t) \right) \]Combine the terms in the numerator:\[ \frac{1}{\cos(t)} \left( \frac{1 - \cos^2(t)}{\cos(t)} \right) \]Multiply out:\[ = \frac{1 - \cos^2(t)}{\cos^2(t)} \]### Step 3: Use a Pythagorean IdentityRecall the Pythagorean identity:\[ \sin^2(t) + \cos^2(t) = 1 \]So,\[ 1 - \cos^2(t) = \sin^2(t) \]Substitute \(\sin^2(t)\) into the expression:\[ \frac{\sin^2(t)}{\cos^2(t)} \]### Step 4: Simplify FurtherRecognize that:\[ \frac{\sin^2(t)}{\cos^2(t)} = \tan^2(t) \]Since the given expression simplifies to \( \sin^2(t) \), the identity is verified:\[ \boxed{\sin^2(t)} \]By carefully substituting identities and simplifying step-by-step, we verify that:\[ \frac{\sec(t) - \cos(t)}{\sec(t)}

Answered: Verify the identity.

Verify the identity.

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

Similar questions

If A - B and B = C, then A - C. %3D True False
Perform the operation
2- evx 3Vx e әх хр
43y It ex
simpplify i27
I have to write 0 in the form of a/b
Evaluate v-4 Enter DNE if not a real number.
ASSESS Learning Activity 5 True or False! Write TRUE if the statement is true. Otherwise, write FALSE. 1. The equation of the circle is derived using the distance formula. 2. The length of the radius can be obtained by getting the distance between a point on the circle and the center. 3. The length of the radius is always positive. 4. In x2 + y2 = 4, the radius is 4. 5. The equation of the circle with center at the origin and radius is x2 + y? = r². _6. The equation x2 – y2 = 16 represents a circle. %3D 7. The distance between the endpoints of a diameter is half of the circle's radius. 8. The equation x² + y2 + Dx + Ey +F = 0 represents a circle provided that the coefficients of x² and y are not zero.
A and B marked as incorrect