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Math  /  Algebra

QuestionProve that tan⁡x1−tan⁡x+cot⁡x1−cot⁡x=−1\frac{\tan x}{1-\tan x}+\frac{\cot x}{1-\cot x}=-1 If cos⁡x=−3/4\cos x=-3 / 4 and xx is in second quadrant find 3sin⁡x−2cot⁡xtan⁡2x\frac{3 \sin x-2 \cot x}{\tan ^{2} x} Find the sum 3+1+1/3+1/9+……3+1+1 / 3+1 / 9+\ldots \ldots
The number of first-year students enrolled at CBU in 2021 was 1290. If 94%94 \% of the students only proceeded and enrolled in the following academic year, find out how many will be enrolled in the 5th 5^{\text {th }} year. Solve the radical equation x−1+3x=0\sqrt{x-1}+\sqrt{3 x}=0 Sketch the graph of y=2+3sin⁡(x)y=2+3 \sin (x) for one cycle Find the angle subtended at the centre OO, if the sector area is 500 cm2500 \mathrm{~cm}^{2} and radius is 10 cm . Solve the logarithmic equation ln⁡x−ln⁡(x−1)=ln⁡(2x2)\ln x-\ln (x-1)=\ln \left(2 x^{2}\right) Solve the equation 3x+1=2x−13^{x+1}=2^{x-1} If e3x+2=5e^{3 x+2}=5, find xx

Studdy Solution

STEP 1

**
1. We need to utilize trigonometric identities related to tangent and cotangent.
2. tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x} and cot⁡x=cos⁡xsin⁡x\cot x = \frac{\cos x}{\sin x}.
3. Simplifying and combining fractions will help in proving the identity.
**

STEP 2

1. **
2. Express tan⁡x\tan x and cot⁡x\cot x in terms of sin⁡x\sin x and cos⁡x\cos x.
3. Simplify each fraction separately.
4. Combine the two fractions and simplify to show that they equal −1-1.
**

STEP 3

** Express tan⁡x\tan x and cot⁡x\cot x in terms of sin⁡x\sin x and cos⁡x\cos x.
tan⁡x1−tan⁡x=sin⁡xcos⁡x1−sin⁡xcos⁡x=sin⁡xcos⁡x−sin⁡x \frac{\tan x}{1 - \tan x} = \frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x}} = \frac{\sin x}{\cos x - \sin x}
cot⁡x1−cot⁡x=cos⁡xsin⁡x1−cos⁡xsin⁡x=cos⁡xsin⁡x−cos⁡x \frac{\cot x}{1 - \cot x} = \frac{\frac{\cos x}{\sin x}}{1 - \frac{\cos x}{\sin x}} = \frac{\cos x}{\sin x - \cos x}
**

STEP 4

** Simplify each fraction separately.
tan⁡x1−tan⁡x=sin⁡xcos⁡x−sin⁡x \frac{\tan x}{1 - \tan x} = \frac{\sin x}{\cos x - \sin x}
cot⁡x1−cot⁡x=cos⁡xsin⁡x−cos⁡x \frac{\cot x}{1 - \cot x} = \frac{\cos x}{\sin x - \cos x}
Notice that sin⁡x−cos⁡x=−(cos⁡x−sin⁡x)\sin x - \cos x = -(\cos x - \sin x).
**

STEP 5

** Combine the two fractions and simplify.
sin⁡xcos⁡x−sin⁡x+cos⁡xsin⁡x−cos⁡x=sin⁡xcos⁡x−sin⁡x+cos⁡x−(cos⁡x−sin⁡x) \frac{\sin x}{\cos x - \sin x} + \frac{\cos x}{\sin x - \cos x} = \frac{\sin x}{\cos x - \sin x} + \frac{\cos x}{-(\cos x - \sin x)}
=sin⁡xcos⁡x−sin⁡x−cos⁡xcos⁡x−sin⁡x = \frac{\sin x}{\cos x - \sin x} - \frac{\cos x}{\cos x - \sin x}
=sin⁡x−cos⁡xcos⁡x−sin⁡x = \frac{\sin x - \cos x}{\cos x - \sin x}
=−1 = -1
Therefore, we have:
tan⁡x1−tan⁡x+cot⁡x1−cot⁡x=−1 \frac{\tan x}{1 - \tan x} + \frac{\cot x}{1 - \cot x} = -1

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