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Difficulty: HardFundamental Trigonometric Identities

An angle θ\theta is positioned in the second quadrant. If the trigonometric expression csc⁡θ−cot⁡θ\csc\theta - \cot\theta is equal to 44, what is the value of 17cos⁡θ17\cos\theta?

Answer: -15

Answer

The value of 17cos⁡θ17\cos\theta is −15-15.
The correct value is −15-15. By using the difference of squares on the identity csc⁡2θ−cot⁡2θ=1\csc^2\theta - \cot^2\theta = 1, we establish that csc⁡θ+cot⁡θ=14\csc\theta + \cot\theta = \frac{1}{4}. Solving this system alongside csc⁡θ−cot⁡θ=4\csc\theta - \cot\theta = 4 yields csc⁡θ=178\csc\theta = \frac{17}{8} and cot⁡θ=−158\cot\theta = -\frac{15}{8}. Since cos⁡θ=cot⁡θcsc⁡θ\cos\theta = \frac{\cot\theta}{\csc\theta}, we find cos⁡θ=−1517\cos\theta = -\frac{15}{17}, which means 17cos⁡θ=−1517\cos\theta = -15.

Step-by-Step Solution

1
Use the Pythagorean identity linking cosecant and cotangent to find the sum of the terms.
csc⁡θ+cot⁡θ=14\csc\theta + \cot\theta = \frac{1}{4}
Since csc⁡2θ−cot⁡2θ=1\csc^2\theta - \cot^2\theta = 1, factoring as a difference of squares gives (csc⁡θ−cot⁡θ)(csc⁡θ+cot⁡θ)=1(\csc\theta - \cot\theta)(\csc\theta + \cot\theta) = 1. Substituting csc⁡θ−cot⁡θ=4\csc\theta - \cot\theta = 4 yields 4(csc⁡θ+cot⁡θ)=14(\csc\theta + \cot\theta) = 1, so csc⁡θ+cot⁡θ=14\csc\theta + \cot\theta = \frac{1}{4}.
2
Solve the system of equations for csc⁡θ\csc\theta and cot⁡θ\cot\theta.
csc⁡θ=178\csc\theta = \frac{17}{8} and cot⁡θ=−158\cot\theta = -\frac{15}{8}
Adding the equations csc⁡θ−cot⁡θ=4\csc\theta - \cot\theta = 4 and csc⁡θ+cot⁡θ=14\csc\theta + \cot\theta = \frac{1}{4} gives 2csc⁡θ=174  ⟹  csc⁡θ=1782\csc\theta = \frac{17}{4} \implies \csc\theta = \frac{17}{8}. Subtracting the first equation from the second gives 2cot⁡θ=−154  ⟹  cot⁡θ=−1582\cot\theta = -\frac{15}{4} \implies \cot\theta = -\frac{15}{8}.
3
Determine the value of cos⁡θ\cos\theta using reciprocal and quotient identities.
cos⁡θ=−1517\cos\theta = -\frac{15}{17}
Using the relationship cos⁡θ=cot⁡θcsc⁡θ\cos\theta = \frac{\cot\theta}{\csc\theta}, we substitute the values to find cos⁡θ=−15/817/8=−1517\cos\theta = \frac{-15/8}{17/8} = -\frac{15}{17}.
4
Calculate the final required expression value.
17cos⁡θ=−1517\cos\theta = -15
Multiplying the calculated value of cos⁡θ\cos\theta by 1717 gives 17(−1517)=−1517 \left(-\frac{15}{17}\right) = -15.

Key Concept

Using fundamental Pythagorean, reciprocal, and quotient trigonometric identities to solve systems of equations and evaluate trigonometric expressions.
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Topics

Fundamental Trigonometric Identities

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