Apply reciprocal, quotient, and Pythagorean identities to simplify complex trigonometric expressions into their most basic forms. This stack focuses on the practical application of identities through algebraic manipulation and substitution.
Use the quotient identity \(\cot(x) = \frac{\cos(x)}{\sin(x)}\).
$$ \sin(x) \cdot \frac{\cos(x)}{\sin(x)} = \cos(x) $$
Simplify:
$$ \frac{\tan(\theta)}{\sec(\theta)} $$
Convert to \(\sin\) and \(\cos\):
$$ \frac{\frac{\sin(\theta)}{\cos(\theta)}}{\frac{1}{\cos(\theta)}} = \frac{\sin(\theta)}{\cos(\theta)} \cdot \frac{\cos(\theta)}{1} = \sin(\theta) $$
Simplify the expression:
$$ \cos^2(y) (1 + \tan^2(y)) $$
Use the Pythagorean identity \(1 + \tan^2(y) = \sec^2(y)\). The expression becomes:
$$ \cos^2(y) \cdot \sec^2(y) = \cos^2(y) \cdot \frac{1}{\cos^2(y)} = 1 $$
Simplify:
$$ \sec^2(\alpha) - 1 $$
Using the Pythagorean identity \(1 + \tan^2(\alpha) = \sec^2(\alpha)\), we can rearrange it to \(\tan^2(\alpha) = \sec^2(\alpha) - 1\). The simplified form is \(\tan^2(\alpha)\).
Simplify:
$$ (1 - \sin^2(x))(\csc^2(x)) $$
Use the Pythagorean identity \(\cos^2(x) = 1 - \sin^2(x)\). The expression becomes:
$$ \cos^2(x) \csc^2(x) = \cos^2(x) \cdot \frac{1}{\sin^2(x)} = \frac{\cos^2(x)}{\sin^2(x)} = \cot^2(x) $$
Simplify the expression:
$$ \frac{\csc^2(x) - 1}{\cos^2(x)} $$
The numerator \(\csc^2(x) - 1 = \cot^2(x)\). The expression becomes:
$$ \frac{\cot^2(x)}{\cos^2(x)} = \frac{\cos^2(x)/\sin^2(x)}{\cos^2(x)} = \frac{1}{\sin^2(x)} = \csc^2(x) $$
Simplify to a single trigonometric function:
$$ \frac{\sin(x) + \tan(x)}{1 + \sec(x)} $$
