Prove the identity ( heta)/(1- heta) + rac heta1- heta = heta + an heta

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ScanSolve AI Tutor · Accepted Answer

To prove the identity f(θ) ≡ cos θ + sin θ, we start with the given expression for f(θ) and simplify it. Given: 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). 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 2: Simplify the denominators of each fraction. For the first term: 1−(sin θ)/(cos θ) = (cos θ)/(cos θ) − (sin θ)/(cos θ) = (cos θ − sin θ)/(cos θ) For the second term: 1−(cos θ)/(sin θ) = (sin θ)/(sin θ) − (cos θ)/(sin θ) = (sin θ − cos θ)/(sin θ) Step 3: Substitute the simplified denominators back into f(θ). f(θ) = (cos θ)/((cos θ − sin θ)/(cos θ)) + (sin θ)/((sin θ − cos θ)/(sin θ)) Step 4: Invert and multiply to simplify the complex fractions. 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 5: Make the denominators common. Notice that sin θ − cos θ = −(cos θ − sin θ). Rewrite the second term: (sin² θ)/(sin θ − cos θ) = (sin² θ)/(−(cos θ − sin θ)) = −(sin² θ)/(cos θ − sin θ) Now substitute this back into the expression for f(θ): f(θ) = (cos² θ)/(cos θ − sin θ) − (sin² θ)/(cos θ − sin θ) Step 6: Combine the fractions. f(θ) = (cos² θ − sin² θ)/(cos θ − sin θ); Step 7: Use the difference of squares identity, a² − b² = (a−b)(a+b). Here, a = cos θ and b = sin θ. So, cos² θ − sin² θ = (cos θ − sin θ)(cos θ + sin θ). Substitute this into the numerator: f(θ) = ((cos θ − sin θ)(cos θ + sin θ))/(cos θ − sin θ) Step 8: Cancel out the common term (cos θ − sin θ) from the numerator and denominator (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 ( heta)/(1- heta) + rac heta1− heta = heta + an heta

Answer

Step-by-step solution

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

Simplify the denominators of each fraction.

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

Invert and multiply to simplify the complex fractions.

Make the denominators common.

Combine the fractions.

Use the difference of squares identity, $a^2 - b^2 = (a-b)(a+b)$ .

Cancel out the common term $(\cos \theta - \sin \theta)$ from the numerator and denominator (assuming $\cos \theta - \sin \theta \neq 0$ ).

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