Solve \frac{\tan\left(\theta \right)}{\sec\left(\theta \right)}=\sin\left(\theta \right) - UpStudy

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Home Math Solver

tanθsecθ=sinθ

error msg

Differential
Integral
Trigonometry
Letters
Algebra
Calculus
Trigonometry
Matrix

Question

\frac{t a n ( θ )}{s e c ( θ )} = sin (θ)

Solve the equation

θ \neq = \frac{π}{2} + kπ, k \in Z

Alternative Form

θ \neq = 9 0^{\circ} + 18 0^{\circ} k, k \in Z

Evaluate

\frac{t a n ( θ )}{s e c ( θ )} = sin (θ)

Find the domain

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Evaluate

{θ \neq = \frac{π}{2} + kπ, k \in Z sec (θ) \neq = 0

Calculate

{θ \neq = \frac{π}{2} + kπ, k \in Z θ \in R

Find the intersection

θ \neq = \frac{π}{2} + kπ, k \in Z

\frac{t a n ( θ )}{s e c ( θ )} = sin (θ), θ \neq = \frac{π}{2} + kπ, k \in Z

Rewrite the expression

\frac{\frac{s i n ( θ )}{c o s ( θ )}}{\frac{1}{c o s ( θ )}} = sin (θ)

Simplify the expression

\frac{s i n ( θ ) c o s ( θ )}{c o s ( θ )} = sin (θ)

Reduce the fraction

sin (θ) = sin (θ)

The statement is true for any value of θ

θ \in R

Check if the solution is in the defined range

θ \in R, θ \neq = \frac{π}{2} + kπ, k \in Z

Solution

θ \neq = \frac{π}{2} + kπ, k \in Z

Alternative Form

θ \neq = 9 0^{\circ} + 18 0^{\circ} k, k \in Z

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Verify the identity

true

Evaluate

\frac{t a n ( θ )}{s e c ( θ )} = sin (θ)

Start working on the left-hand side

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Evaluate

\frac{t a n ( θ )}{s e c ( θ )}

Use tan t = \frac{s i n t}{c o s t} to transform the expression

\frac{\frac{s i n ( θ )}{c o s ( θ )}}{s e c ( θ )}

Multiply by the reciprocal

\frac{s i n ( θ )}{c o s ( θ )} \times \frac{1}{s e c ( θ )}

Multiply the terms

\frac{s i n ( θ )}{c o s ( θ ) s e c ( θ )}

Transform the expression

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Evaluate

cos (θ) sec (θ)

Use sec t = \frac{1}{c o s t} to transform the expression

cos (θ) \times \frac{1}{c o s ( θ )}

Cancel out the common factor cos (θ)

1 \times 1

Multiply the terms

1

\frac{s i n ( θ )}{1}

Divide the terms

sin (θ)

sin (θ) = sin (θ)

Solution

true

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