Is cos^2(θ)(tan^2(θ)+1)=1 ?

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

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prove cos^2(θ)(tan^2(θ)+1)=1

prove

cos^{2} (θ) (tan^{2} (θ) + 1) = 1

True

Solution steps

cos^{2} (θ) (tan^{2} (θ) + 1) = 1

Manipulating left side

cos^{2} (θ) (tan^{2} (θ) + 1)

Express with sin, cos

(1 + tan^{2} (θ)) cos^{2} (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Join

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

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

Convert element to fraction:

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

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

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

Multiply:

1 \cdot cos^{2} (θ) = cos^{2} (θ)

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (θ)

= cos^{2} (θ) + sin^{2} (θ)

= cos^{2} (θ) + sin^{2} (θ)

= cos^{2} (θ) + sin^{2} (θ)

Rewrite using trig identities

cos^{2} (θ) + sin^{2} (θ)

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

sec^{2} (\frac{θ}{2}) = \frac{2}{1 + cos ( θ )}

\frac{1}{sin ( x ) + 1} + \frac{1}{csc ( x ) + 1} = 1

sin^{2} (θ) - cos^{2} (θ) = 1 - 2 cos^{2} (θ)

sin (t) csc (t) = 1

csc (x) (1 - cos^{2} (x)) = sin (x)

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