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Chapter 19: Problem 1854

if \(\tan \theta+a b \cot \theta=a+b\) then \(\tan \theta\) (a) a (b) \(\mathrm{b}\) (c) a or \(\mathrm{b}\) (d) \((\pi / 4)\)

Short Answer

Expert verified
The short answer is: \(\tan \theta\) = a or b (Option (c)).

Step by step solution

01

Identify the given equation

Given that \(\tan \theta + ab \cot \theta = a + b\).
02

Rewrite cotangent function in terms of tangent function

We know that \(\cot \theta = \frac{1}{\tan \theta}\). We can substitute this expression into the given equation: $$\tan \theta + ab \left(\frac{1}{\tan \theta}\right) = a + b$$
03

Eliminate the fraction

To eliminate the fraction, we can multiply the entire equation by \(\tan \theta\): $$\tan^2 \theta + ab = (a + b) \tan \theta$$
04

Rearrange the equation to form a quadratic equation in terms of tanθ

Rearrange the given equation to form a quadratic equation in terms of \(\tan \theta\): $$\tan^2 \theta - (a + b) \tan \theta + ab = 0$$
05

Solve the quadratic equation

Solve the quadratic equation $$\tan^2 \theta - (a + b) \tan \theta + ab = 0$$ for \(\tan \theta\): Since this equation is in the form of ax^2 + bx + c = 0 where, x = \(\tan \theta\), a = 1, b = -(a+b) and c = ab According to the quadratic formula, the solutions are given by: $$\tan \theta = \frac{-(a+b) \pm \sqrt{((a+b)^2) - 4(1)(ab)}}{2(1)}$$
06

Simplify the solution

Simplify the value of \(\tan \theta\): $$\tan \theta = \frac{-(a+b) \pm \sqrt{(a^2+2ab+b^2) - 4ab}}{2}$$ $$\tan \theta = \frac{-(a+b) \pm \sqrt{a^2 - 2ab + b^2}}{2}$$
07

Compare with the given options

Comparing the value of \(\tan \theta\) with the options: (a) a (b) b (c) a or b (d) \((\pi / 4)\) We can see that our simplified solution of \(\tan \theta\) is either a or b. So, the correct answer is: \(\tan \theta\) = a or b (Option (c)).

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Most popular questions from this chapter

If \(2 \sec ^{2} \alpha-\sec ^{4} \alpha-2 \operatorname{cosec}^{2} \alpha+\operatorname{cosec}^{4} \alpha=(15 / 4)\) then \(\tan ^{2} \alpha=\) (a) \((1 / \sqrt{2})\) (b) \((1 / 2)\) (c) \([1 /(2 \sqrt{2})]\) (d) \((1 / 4)\)

If \(\cos (\alpha+\beta)=(4 / 5), \sin (\alpha-\beta)=(5 / 13), 0<\alpha, \beta<(\pi / 4)\) then \(\cot 2 \alpha=\) (a) \((12 / 19)\) (b) \((7 / 20)\) (c) \((16 / 25)\) (d) \((33 / 56)\)

There is a bridge of the length \(h\) on a valley. The angle of depression of a temple lying in a valley from two ends of a bridge are \(\alpha\) and \(\beta\), then the height of the bridge from top of the temple \(=\) (a) \([(h \tan \alpha \tan \beta) /(\tan \alpha-\tan \beta)]\) (b) \([(h \tan \alpha \tan \beta) /(\tan \alpha+\tan \beta)]\) (c) \([(\tan \alpha \tan \beta) /\\{h(\tan \alpha-\tan \beta)\\}]\) (d) \([\\{h(\tan \alpha+\tan \beta)\\} /(\tan \alpha \tan \beta)]\)

\(\sin ^{-1}(\sin 2)+\sin ^{-1}(\sin 4)+\sin ^{-1}(\sin 6)=\) (a) \(\pi-12\) (b) 0 (c) 12 (d) \(12-\pi\)

If \(A=\cos ^{4} \theta+\sin ^{2} \theta, \forall \theta \in R\) then \(A\) lies in the interval (a) \([1,2]\) (b) \([(3 / 4), 1]\) (c) \([(13 / 16), 1]\) (d) \([(3 / 4),(13 / 16)]\)
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