Prove the identity ( theta)/(1- theta) + ( theta)/(1- theta) = theta + theta.

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To prove the identity f(θ) ≡ cos θ + sin θ, we start with the given expression for f(θ): f(θ) = (cos θ)/(1−tan θ) + (sin θ)/(1−cot θ) Step 1: Express tan θ and cot θ in terms of sin θ and cos θ. We know that tan θ = (sin θ)/(cos θ) and cot θ = (cos θ)/(sin θ). Step 2: Substitute these into the expression for f(θ). f(θ) = (cos θ)/(1−(sin θ)/(cos θ)) + (sin θ)/(1−(cos θ)/(sin θ)) Step 3: Simplify the denominators by finding a common denominator for each term. 1−(sin θ)/(cos θ) = (cos θ)/(cos θ) − (sin θ)/(cos θ) = (cos θ − sin θ)/(cos θ); 1−(cos θ)/(sin θ) = (sin θ)/(sin θ) − (cos θ)/(sin θ) = (sin θ − cos θ)/(sin θ) Step 4: Substitute the simplified denominators back into the expression for f(θ). f(θ) = (cos θ)/((cos θ − sin θ)/(cos θ)) + (sin θ)/((sin θ − cos θ)/(sin θ)) Step 5: Invert the denominators and multiply. f(θ) = cos θ · (cos θ)/(cos θ − sin θ) + sin θ · (sin θ)/(sin θ − cos θ); f(θ) = (cos² θ)/(cos θ − sin θ) + (sin² θ)/(sin θ − cos θ); Step 6: Notice that the denominators are negatives of each other: sin θ − cos θ = −(cos θ − sin θ). Rewrite the second term with the common denominator cos θ − sin θ. f(θ) = (cos² θ)/(cos θ − sin θ) + (sin² θ)/(−(cos θ − sin θ)); f(θ) = (cos² θ)/(cos θ − sin θ) − (sin² θ)/(cos θ − sin θ) Step 7: Combine the fractions since they now have a common denominator. f(θ) = (cos² θ − sin² θ)/(cos θ − sin θ); Step 8: Apply the difference of squares identity, a² − b² = (a−b)(a+b), where a = cos θ and b = sin θ. cos² θ − sin² θ = (cos θ − sin θ)(cos θ + sin θ) Step 9: Substitute this back into the expression for f(θ). f(θ) = ((cos θ − sin θ)(cos θ + sin θ))/(cos θ − sin θ) Step 10: Cancel out the common term (cos θ − sin θ), assuming cos θ − sin θ ≠ 0. f(θ) = cos θ + sin θ Thus, we have proven that f(θ) ≡ cos θ + sin θ. The final answer is cos θ + sin θ.

Prove the identity ( theta)/(1- theta) + ( theta)/(1− theta) = theta + theta.

Answer

Step-by-step solution

Express $\tan \theta$ and $\cot \theta$ in terms of $\sin \theta$ and $\cos \theta$ .

Substitute these into the expression for $f(\theta)$ .

Simplify the denominators by finding a common denominator for each term.

Substitute the simplified denominators back into the expression for $f(\theta)$ .

Invert the denominators and multiply.

Notice that the denominators are negatives of each other

Combine the fractions since they now have a common denominator.

Apply the difference of squares identity, $a^2 - b^2 = (a-b)(a+b)$ , where $a = \cos \theta$ and $b = \sin \theta$ .

Substitute this back into the expression for $f(\theta)$ .

Cancel out the common term $(\cos \theta - \sin \theta)$ , assuming $\cos \theta - \sin \theta \neq 0$ .

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