Problem 34
Question
Verify each identity. $$\csc ^{2} x \sec x=\sec x+\csc x \cot x$$
Step-by-Step Solution
Verified Answer
The given trigonometric identity \(\csc ^{2} x \sec x=\sec x+\csc x \cot x\) is verified as true.
1Step 1: Rewrite the left side of the equation
Replace \(\csc ^{2} x\) and \(\sec x\) with their respective expressions in terms of sine, cosine and tangent. This gives us \(\frac{1}{\sin^2{x}} \cdot \frac{1}{\cos{x}}\).
2Step 2: Simplify the expression
Simplify the expression using the property of fractions. This gives us \(\frac{1}{\sin^2 x \cos x}\).
3Step 3: Rewrite in terms of sec and csc
Rearrange the expression in terms of sec and csc to get \(\sec x + \csc x \cot x\). Using the identities: sec x =\(\frac{1}{\cos x}\), csc x = \(\frac{1}{\sin x}\) and cot x = \(\frac{cos x}{sin x}\)
4Step 4: Equate the expressions
Equate the right side and the simplified left side of the equation. If both sides are equivalent, then the given trigonometric identity is verified.
Other exercises in this chapter
Problem 33
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Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$\cos \frac{2 \theta}{3}=-1$$
View solution