Trigonometry Examples

$(1 + \tan^{2} (u)) (1 - \sin^{2} (u)) = 1$

Step 1

Simplify the left side.

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Step 1.1

Simplify $(1 + \tan^{2} (u)) (1 - \sin^{2} (u))$ .

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Step 1.1.1

Rearrange terms.

$(\tan^{2} (u) + 1) (1 - \sin^{2} (u)) = 1$

Step 1.1.2

Apply pythagorean identity.

$\sec^{2} (u) (1 - \sin^{2} (u)) = 1$

Step 1.1.3

Rewrite $\sec (u)$ in terms of sines and cosines.

${(\frac{1}{\cos (u)})}^{2} (1 - \sin^{2} (u)) = 1$

Step 1.1.4

Simplify the expression.

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Step 1.1.4.1

Apply the product rule to $\frac{1}{\cos (u)}$ .

$\frac{1^{2}}{\cos^{2} (u)} (1 - \sin^{2} (u)) = 1$

Step 1.1.4.2

One to any power is one.

$\frac{1}{\cos^{2} (u)} (1 - \sin^{2} (u)) = 1$

Step 1.1.5

Apply pythagorean identity.

$\frac{1}{\cos^{2} (u)} \cos^{2} (u) = 1$

Step 1.1.6

Cancel the common factor of $\cos^{2} (u)$ .

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Step 1.1.6.1

Cancel the common factor.

$\frac{1}{\cos^{2} (u)} \cos^{2} (u) = 1$

Step 1.1.6.2

Rewrite the expression.

$1 = 1$

Step 2

Since $1 = 1$ , the equation will always be true for any value of $u$ .

All real numbers

Step 3

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$