8.4 Trigonometric Identities: The Logic of Equality

Step 1

The square root blocks direct simplification. Strategy C: Rationalize the denominator.

Step 2

Multiply numerator and denominator inside the root by $(1 + \sin A)$ .

Step 3

$\sqrt{\frac{(1 + \sin A)(1 + \sin A)}{(1 - \sin A)(1 + \sin A)}} = \sqrt{\frac{(1 + \sin A)^2}{1 - \sin^2 A}}$ .

Step 4

Use Identity: $1 - \sin^2 A = \cos^2 A$ .

Step 5

Expression becomes $\sqrt{\frac{(1 + \sin A)^2}{\cos^2 A}}$ .

Step 6

Remove square root: $\frac{1 + \sin A}{\cos A}$ .

Step 7

Split the fraction: $\frac{1}{\cos A} + \frac{\sin A}{\cos A}$ .

Step 8

Result: $\sec A + \tan A$ . LHS = RHS.

Final Answer

Proved.

Operation

Angle

Unit

Standard Values:

sin 0 = 0, sin 30 = 0.5, sin 45 = 0.707, sin 60 = 0.866, sin 90 = 1

cos 0 = 1, cos 30 = 0.866, cos 45 = 0.707, cos 60 = 0.5, cos 90 = 0

Select an operation and enter values.