If θ = 12/13 then what is the value of ( θ + θ)² ( θ - θ)^{-2} 0

Correct Answer is : 2254225/44225

Correct Answer is : 2254225/44225

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UPSC CDS 1 2023 Math Paper

Question : 47 of 100

Marks: +1, -0

\sin \theta = \frac{12}{13}

(\tan \theta + \sec \theta)^{2} (\csc \theta - \cot \theta)^{-2}

0 < \theta < \frac{\pi}{2}

Solution:

Concept:

Use trigonometric ratios and the Pythagorean theorem to find all six ratios from

\sin \theta = \frac{12}{13}

for an acute angle.

Explanation:

Given

\sin \theta = \frac{12}{13} = \frac{\text{opposite}}{\text{hypotenuse}}

Let opposite side

= 12

, hypotenuse

= 13

Using

(\text{hypotenuse})^2 = (\text{opposite})^2 + (\text{adjacent})^2

13^2 = 12^2 + (\text{adjacent})^2 \Rightarrow 169 = 144 + (\text{adjacent})^2

(\text{adjacent})^2 = 25 \Rightarrow \text{adjacent} = 5

Now compute the needed ratios:

\tan \theta = \frac{12}{5}

\sec \theta = \frac{13}{5}

\csc \theta = \frac{13}{12}

\cot \theta = \frac{5}{12}

First part:

\tan \theta + \sec \theta = \frac{12}{5} + \frac{13}{5} = \frac{25}{5} = 5

(\tan \theta + \sec \theta)^2 = 5^2 = 25

Second part:

\csc \theta - \cot \theta = \frac{13}{12} - \frac{5}{12} = \frac{8}{12} = \frac{2}{3}

Then

(\csc \theta - \cot \theta)^{-2} = \left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}

Multiply:

25 \times \frac{9}{4} = \frac{225}{4}

Answer:

\frac{225}{4}

(Option D).

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