Questions: Simplify and write the trigonometric expression in terms of sine and cosine: (1+cos y)/(1+sec y)=□

Simplify and write the trigonometric expression in terms of sine and cosine:
(1+cos y)/(1+sec y)=□

Transcript text: Assignment 7.1: Solving Trigonometric Equations with Identities (Required) Assignment 7.1: Solving Trigonometric Equations with Score: 11/13 Answered: 11/13 Question 12 Simplify and write the trigonometric expression in terms of sine and cosine: \[ \frac{1+\cos y}{1+\sec y}=\square \] Question Help: Video Submit Question

Solution

Solution Steps

Step 1: Rewrite \(\sec y\) in terms of cosine

\[ \sec y = \frac{1}{\cos y} \]

Step 2: Substitute \(\sec y\) into the expression

\[ \frac{1 + \cos y}{1 + \sec y} = \frac{1 + \cos y}{1 + \frac{1}{\cos y}} \]

Step 3: Simplify the denominator

\[ 1 + \frac{1}{\cos y} = \frac{\cos y + 1}{\cos y} \]

Step 4: Rewrite the original expression with the simplified denominator

\[ \frac{1 + \cos y}{\frac{\cos y + 1}{\cos y}} = (1 + \cos y) \cdot \frac{\cos y}{\cos y + 1} \]

Step 5: Cancel out \((1 + \cos y)\) in the numerator and denominator

\[ (1 + \cos y) \cdot \frac{\cos y}{\cos y + 1} = \cos y \]

The expression simplifies to \(\cos y\).

Final Answer

\(\boxed{\cos y}\)

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