Problem 11 Verify the identity. $$\cot ^{... [FREE SOLUTION]

Chapter 5: Problem 11

Verify the identity. $$\cot ^{2} y\left(\sec ^{2} y-1\right)=1$$

Short Answer

Expert verified

The trigonometric identity $\cot^{2}(y)( \sec^{2}(y) - 1) = 1$ holds true for all values of $y$, for which $\cot y$ and $\sec y$ are defined and $y 鈮� n\pi$, where $n$ is not a zero integer, as here the sine function in the denominator would make it undefined.

Step by step solution

Express cot^2(y) in terms of cosine and sine

The cotangent function, $\cot(y)$, is the reciprocal of the tangent function. This means $\cot(y) = \frac{\cos y}{\sin y}$. Therefore, $\cot^2(y) = \frac{\cos^2 y}{\sin^2 y}$

Express sec^2(y) - 1 in terms of cosine and sine

The secant function, $\sec(y)$, is the reciprocal of the cosine function. This means $\sec(y) = \frac{1}{\cos y}$. Therefore, $\sec^2(y) - 1 = \frac{1}{\cos^2 y} - 1 = \frac{1 - \cos^2 y}{\cos^2 y} = \frac{\sin^2 y}{\cos^2 y}$

Substitute cot^2(y) and sec^2(y) - 1 into the originnal equation

Now, replace $\cot^2(y) = \frac{\cos^2 y}{\sin^2 y}$ and $\sec^2(y) - 1 = \frac{\sin^2 y}{\cos^2 y}$ in the original equation. It thus becomes $\frac{\cos^2 y}{\sin^2 y} * \frac{\sin^2 y}{\cos^2 y} = 1$

Simplify the result

By cancelling out the identical terms from the numerator and the denominator in both functions, the left-hand side will now become 1. Consequently, the verification of the initial identity is completed: $1 = 1$

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91影视

Verify the identity. $$\cot ^{2} y\left(\sec ^{2} y-1\right)=1$$

Short Answer

Step by step solution

Express cot^2(y) in terms of cosine and sine

Express sec^2(y) - 1 in terms of cosine and sine

Substitute cot^2(y) and sec^2(y) - 1 into the originnal equation

Simplify the result

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