Is sin(x)+sin(x)tan^2(x)=tan(x)sec(x) ?

The answer to whether sin(x)+sin(x)tan^2(x)=tan(x)sec(x) is True

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prove sin(x)+sin(x)tan^2(x)=tan(x)sec(x)

Solution

prove

sin (x) + sin (x) tan^{2} (x) = tan (x) sec (x)

Solution

True

Solution steps

sin (x) + sin (x) tan^{2} (x) = tan (x) sec (x)

Manipulating left side

sin (x) + sin (x) tan^{2} (x)

Factor

sin (x) + sin (x) tan^{2} (x) : sin (x) (1 + tan^{2} (x))

sin (x) + sin (x) tan^{2} (x)

Factor out common term

sin (x)

= sin (x) (1 + tan^{2} (x))

= (1 + tan^{2} (x)) sin (x)

Express with sin, cos

(1 + tan^{2} (x)) sin (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= (1 + (\frac{sin ( x )}{cos ( x )})^{2}) sin (x)

Simplify

(1 + (\frac{sin ( x )}{cos ( x )})^{2}) sin (x) : \frac{sin ( x ) ( cos ^{2} ( x ) + sin ^{2} ( x ) )}{cos ^{2} ( x )}

(1 + (\frac{sin ( x )}{cos ( x )})^{2}) sin (x)

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= sin (x) (\frac{sin ^{2} ( x )}{cos ^{2} ( x )} + 1)

Join

1 + \frac{sin ^{2} ( x )}{cos ^{2} ( x )} : \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )}

1 + \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

Convert element to fraction:

1 = \frac{1 cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{1 \cdot cos ^{2} ( x )}{cos ^{2} ( x )} + \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )}

Multiply:

1 \cdot cos^{2} (x) = cos^{2} (x)

= \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )} sin (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( cos ^{2} ( x ) + sin ^{2} ( x ) ) sin ( x )}{cos ^{2} ( x )}

= \frac{sin ( x ) ( cos ^{2} ( x ) + sin ^{2} ( x ) )}{cos ^{2} ( x )}

= \frac{( cos ^{2} ( x ) + sin ^{2} ( x ) ) sin ( x )}{cos ^{2} ( x )}

Rewrite using trig identities

\frac{( cos ^{2} ( x ) + sin ^{2} ( x ) ) sin ( x )}{cos ^{2} ( x )}

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

= \frac{1 \cdot sin ( x )}{cos ^{2} ( x )}

Simplify

= \frac{sin ( x )}{cos ^{2} ( x )}

= \frac{sin ( x )}{cos ^{2} ( x )}

Manipulating right side

tan (x) sec (x)

Express with sin, cos

sec (x) tan (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( x )} tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1}{cos ( x )} \cdot \frac{sin ( x )}{cos ( x )}

Simplify

\frac{1}{cos ( x )} \cdot \frac{sin ( x )}{cos ( x )} : \frac{sin ( x )}{cos ^{2} ( x )}

\frac{1}{cos ( x )} \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot sin ( x )}{cos ( x ) cos ( x )}

Multiply:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x )}{cos ( x ) cos ( x )}

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

= \frac{sin ( x )}{cos ^{2} ( x )}

= \frac{sin ( x )}{cos ^{2} ( x )}

= \frac{sin ( x )}{cos ^{2} ( x )}

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

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sec (y) cos (y) = 1

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Frequently Asked Questions (FAQ)

Is sin(x)+sin(x)tan^2(x)=tan(x)sec(x) ?
The answer to whether sin(x)+sin(x)tan^2(x)=tan(x)sec(x) is True