Is (1+tan^2(α))cos^2(α)=1 ?

The answer to whether (1+tan^2(α))cos^2(α)=1 is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

prove (1+tan^2(α))cos^2(α)=1

prove

(1 + tan^{2} (α)) cos^{2} (α) = 1

True

Solution steps

(1 + tan^{2} (α)) cos^{2} (α) = 1

Manipulating left side

(1 + tan^{2} (α)) cos^{2} (α)

Express with sin, cos

(1 + tan^{2} (α)) cos^{2} (α)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Join

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

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

Convert element to fraction:

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

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

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} ( α ) + sin ^{2} ( α )}{cos ^{2} ( α )}

Multiply:

1 \cdot cos^{2} (α) = cos^{2} (α)

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (α)

= cos^{2} (α) + sin^{2} (α)

= cos^{2} (α) + sin^{2} (α)

= cos^{2} (α) + sin^{2} (α)

Rewrite using trig identities

cos^{2} (α) + sin^{2} (α)

Use the Pythagorean identity:

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

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

p ro v e \frac{1}{sin ^{2} ( x )} = \frac{1}{csc ^{2} ( x )}

p ro v e sin (x) (1 + tan^{2} (x)) = sin (x) sec^{2} (x)

p ro v e \frac{tan ^{2} ( X )}{1 + cot ^{2} ( X )} = sin^{4} (X)

p ro v e sin (x) + cot (x) cos (x) = \frac{1}{sin ( x )}

p ro v e 1 + cos (6 x) = 2 cos^{2} (3 x)

Is (1+tan^2(α))cos^2(α)=1 ?
The answer to whether (1+tan^2(α))cos^2(α)=1 is True