Perform each indicated operation and simplify the result so that there are no quotients.
cos β(sec β + csc β)

Guida verificata passo dopo passo

Start by writing down the given expression: \(\cos \beta (\sec \beta + \csc \beta)\).

Recall the definitions of the secant and cosecant functions in terms of cosine and sine: \(\sec \beta = \frac{1}{\cos \beta}\) and \(\csc \beta = \frac{1}{\sin \beta}\).

Substitute these definitions into the expression to get: \(\cos \beta \left( \frac{1}{\cos \beta} + \frac{1}{\sin \beta} \right)\).

Distribute \(\cos \beta\) across the terms inside the parentheses: \(\cos \beta \cdot \frac{1}{\cos \beta} + \cos \beta \cdot \frac{1}{\sin \beta}\).

Simplify each term: the first term simplifies to 1, and the second term becomes \(\frac{\cos \beta}{\sin \beta}\). Recognize that \(\frac{\cos \beta}{\sin \beta}\) is \(\cot \beta\). So the simplified expression is \(1 + \cot \beta\).

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Reciprocal Trigonometric Functions

Secant (sec) and cosecant (csc) are reciprocal functions of cosine and sine, respectively. Specifically, sec β = 1/cos β and csc β = 1/sin β. Understanding these relationships helps in rewriting expressions to eliminate quotients.

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The distributive property allows multiplication of a single term across terms inside parentheses. For example, cos β(sec β + csc β) expands to cos β·sec β + cos β·csc β, enabling simplification by applying trigonometric identities.

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