Problem 766 Show that \(\tan t+\cot t=\csc t... [FREE SOLUTION]

Chapter 35: Problem 766

Show that \(\tan t+\cot t=\csc t \sec t\)

Short Answer

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We used trigonometric identities to rewrite the original equation \(\tan t + \cot t = \csc t \cdot \sec t\) in terms of sin and cos functions. After simplifying and finding a common denominator, we showed that both sides are equal, proving the given identity: \[\frac{\sin^2 t + \cos^2 t}{\sin t \cos t} = \frac{1}{\sin t \cos t} = \frac{1}{\sin t} \cdot \frac{1}{\cos t}\]

Step by step solution

Write the given expressions in terms of basic trigonometric functions

To start, let's rewrite \(\tan t\), \(\cot t\), \(\csc t\), and \(\sec t\) in terms of \(\sin t\) and \(\cos t\) using the trigonometric identities mentioned earlier: - \(\tan t = \frac{\sin t}{\cos t}\) - \(\cot t = \frac{\cos t}{\sin t}\) - \(\csc t = \frac{1}{\sin t}\) - \(\sec t = \frac{1}{\cos t}\) Now, let's substitute these expressions into the original equation: \[ \frac{\sin t}{\cos t} + \frac{\cos t}{\sin t} = \frac{1}{\sin t} \cdot \frac{1}{\cos t} \]

Simplify the left side of the equation

We need to find the common denominator for the sum on the left side, which is \(\sin(t)\cos(t)\). Let's rewrite the left side with a common denominator and simplify: \[ \frac{\sin^2 t}{\sin t \cos t} + \frac{\cos^2 t}{\sin t \cos t} = \frac{\sin^2 t + \cos^2 t}{\sin t \cos t} \] Since \(\sin^2 t + \cos^2 t = 1\), we can simplify this expression further: \[ \frac{1}{\sin t \cos t} \]

Check if both sides are equal

Now, let's compare both sides of the equation: \[ \frac{1}{\sin t \cos t} = \frac{1}{\sin t} \cdot \frac{1}{\cos t} \] We can see that both expressions are equal, which means that our original expression holds true: \[ \tan t + \cot t = \csc t \cdot \sec t \] Therefore, we have shown that \(\tan t + \cot t = \csc t \cdot \sec t\) is a valid identity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tan Function

The tangent function, often abbreviated as "tan," is one of the core trigonometric functions. It is defined by the ratio of the opposite side to the adjacent side in a right triangle. However, in terms of a unit circle perspective, it can be thought of as:

tan(t) = \( \frac{\sin t}{\cos t} \)
It becomes undefined when \(\cos t = 0\).

Its periodic nature repeats every \(180^\circ\) or \(\pi\) radians. Since it involves both the sine and cosine functions, the tan function reflects their changes and zeros. Understanding this function is essential for solving trigonometric identities and equations, as it often appears in ratios and can be rewritten using other trigonometric identities.

Cotangent

Cotangent, or "cot," is another trigonometric function. It is the reciprocal of the tangent function and relates to a different pair of triangle sides:

cot(t) = \( \frac{\cos t}{\sin t} \)
It becomes undefined when \(\sin t = 0\).

Cotangent is also periodic, with a period of \(\pi\), similar to the tangent function but it is the inverse of the ratio that defines tan.
Understanding cotangent tricks is useful for problem-solving in trigonometry; it allows for simplification and transformation of trigonometric equations and can also help with solving identities, like in the original problem where tan and cot are added together.

Cosecant

Cosecant, abbreviated as "csc," is the reciprocal of the sine function. It's not as frequently used, but it plays a crucial role in various trigonometric identities:

csc(t) = \( \frac{1}{\sin t} \)
Undefined at points where \(\sin t = 0\), such as multiples of \(\pi\).

Cosecant functions have vertical asymptotes where sine is zero, creating a series of "gaps" in its graph. Recognizing these properties helps understand how it fits into complex identities, such as converting addition of tan and cot into a product of csc and sec.

Secant

The secant function, often notated as "sec," is the reciprocal of the cosine function. It is most notable for its usefulness in various equations and its relationship to other trigonometric functions:

sec(t) = \( \frac{1}{\cos t} \)
Undefined when \(\cos t = 0\), at odd multiples of \(\frac{\pi}{2}\).

Secant helps to see how trigonometric equations can reflect or transform. In the given identity \(\tan t + \cot t = \csc t \sec t\), secant plays an important role by showing how expressions multiply to give a simple reciprocal identity over sine-cosine products.
Gaining a deep understanding of these reciprocals and their properties not only aids in solving identities but also provides insight into more complicated trigonometric forms.

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91影视

Show that \(\tan t+\cot t=\csc t \sec t\)

Short Answer

Step by step solution

Write the given expressions in terms of basic trigonometric functions

Simplify the left side of the equation

Check if both sides are equal

Key Concepts

Tan Function

Cotangent

Cosecant

Secant

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