Solve for x sin(x)sec(x)cot(x)=1

Problem

sin(x)*sec(x)*cot(x)=1

Solution

1. Express in terms of sine and cosine by using the reciprocal identity sec(x)=1/cos(x) and the quotient identity cot(x)=cos(x)/sin(x)

sin(x)⋅1/cos(x)⋅cos(x)/sin(x)=1

2. Simplify the expression by canceling the common factors sin(x) and cos(x) in the numerator and denominator.

1=1

3. Identify the domain constraints because the original expression contains sec(x) and cot(x) which are undefined when cos(x)=0 or sin(x)=0

cos(x)≠0⇒x≠π/2+n*π

sin(x)≠0⇒x≠n*π

4. Combine the constraints to determine the values of x for which the equation holds true. The equation is an identity for all x where the functions are defined.

x≠(n*π)/2

Final Answer

sin(x)*sec(x)*cot(x)=1⇒x∈ℝ,x≠(n*π)/2* for *n∈ℤ

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