Simplifying the Trigonometric Expression
The question asks to simplify the given trigonometric expression: $ \frac{\sin A \tan A}{(1 - \cos A)} $ We need to find which of the options equals this simplified form.
Step-by-Step Solution
- Step 1: Express tan A Rewrite $ \tan A $ using the fundamental identity $ \tan A = \frac{\sin A}{\cos A} $.
- Step 2: Substitute and Combine Terms Substitute $ \frac{\sin A}{\cos A} $ into the expression: $ \frac{\sin A \left( \frac{\sin A}{\cos A} \right)}{(1 - \cos A)} $ Simplify the numerator: $ \frac{\frac{\sin^2 A}{\cos A}}{(1 - \cos A)} $ Combine into a single fraction: $ \frac{\sin^2 A}{\cos A (1 - \cos A)} $
- Step 3: Use Pythagorean Identity Apply the Pythagorean identity $ \sin^2 A = 1 - \cos^2 A $: $ \frac{1 - \cos^2 A}{\cos A (1 - \cos A)} $
- Step 4: Factor and Cancel Factor the numerator $ (1 - \cos^2 A) $ as a difference of squares ($ (1 - \cos A)(1 + \cos A) $): $ \frac{(1 - \cos A)(1 + \cos A)}{\cos A (1 - \cos A)} $ Cancel the common factor $ (1 - \cos A) $, assuming $ \cos A \neq 1 $: $ \frac{1 + \cos A}{\cos A} $
- Step 5: Final Simplification Split the fraction into two terms: $ \frac{1}{\cos A} + \frac{\cos A}{\cos A} $ Use the identity $ \frac{1}{\cos A} = \sec A $: $ \sec A + 1 $
Conclusion
The simplified expression is $ 1 + \sec A $. This corresponds to the first option.
Similar Questions
- If $\cot A = \sqrt{5}$, then find the value of $\frac{\csc^2 A - \sec^2 A}{\csc^2 A + \sec^2 A}$
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Important Questions from Trigonometric Ratios and Identities
If cosec θ = 13/12, then sin θ + cos θ - tan θ is equal to:
What is the value of \(\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}\) ?
If \(\frac{{\tan \theta + \sin \theta }}{{\tan \theta - \sin \theta }} = \frac{{k + 1}}{{k - 1}},\) then k = ?
If α + β = 90° and α = 2β, then the value of 3 cos 2 α - 2 sin 2 β is equal to:
If \(\sqrt{3}\) tan θ = 3 sin θ, then what is the value of sin 2θ − cos 2θ ?
