Problem 178 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Prove that:- i. \(\quad \sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{8}{17}=\sin ^{-1} \frac{77}{85}\). ii. \(\quad \sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{15}{17}=\pi-\sin ^{-1} \frac{77}{85}\). iii. \(\quad \sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{7}{25}=\cos ^{-1} \frac{253}{325}\). iv. \(\quad \cos ^{-1} \frac{4}{5}+\tan ^{-1} \frac{3}{5}=\tan ^{-1} \frac{27}{11}\). v. \(\quad \cos ^{-1} \frac{4}{5}+\cos ^{-1} \frac{12}{13}=\cos ^{-1} \frac{33}{65}\). vi. \(\quad 2 \cos ^{-1} \frac{3}{\sqrt{13}}+\cot ^{-1} \frac{16}{63}+\frac{1}{2} \cos ^{-1} \frac{7}{25}=\pi\). vii. \(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}=45^{\circ}\). viii. \(\sin ^{-1} \frac{1}{\sqrt{5}}+\cot ^{-1} 3=45^{\circ}\). ix. \(\quad 2 \tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)=\frac{\pi}{4}\). x. \(\quad \tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{13}=\tan ^{-1} \frac{2}{9}\). xi. \(\tan ^{-1} \frac{2}{3}=\frac{1}{2} \tan ^{-1} \frac{12}{5}\). xii. \(\quad \tan ^{-1} \frac{1}{4}+\tan ^{-1} \frac{2}{9}=\frac{1}{2} \cos ^{-1} \frac{3}{5}\). \begin{aligned} &\text { xiii. } \cos ^{-1}\left(\frac{15}{17}\right)+2 \tan ^{-1}\left(\frac{1}{5}\right)=\tan ^{-1} \frac{171}{140} \\ &\text { xiv. } 2 \tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{7}+2 \tan ^{-1} \frac{1}{8}=\frac{\pi}{4} . \\ &\text { xv. } \tan ^{-1} \frac{3}{4}+\tan ^{-1} \frac{3}{5}-\tan ^{-1} \frac{8}{19}=\frac{\pi}{4} \\ &\text { xvi. } \tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{8}=\frac{\pi}{4} \\ &\text { xvii. } 3 \tan ^{-1} \frac{1}{4}+\tan ^{-1} \frac{1}{20}=\frac{\pi}{4}-\tan ^{-1} \frac{1}{1985} . \\ &\text { xviii. } 4 \tan ^{-1} \frac{1}{5}-\tan ^{-1} \frac{1}{70}+\tan ^{-1} \frac{1}{99}=\frac{\pi}{4} . \\ &\text { xix. } \sin ^{-1} \frac{3}{5}+\cos e c^{-1} \frac{5}{4}=\frac{\pi}{2} . \\ &\text { xx. } \tan ^{-1} \frac{120}{119}=2 \sin ^{-1} \frac{5}{13} \text { . } \\ &\text { xxi. } \cos \left(2 \tan ^{-1} \frac{1}{7}\right)=\sin \left(4 \tan ^{-1} \frac{1}{3}\right) . \\ &\text { xxii. } \tan ^{-1} t+\tan ^{-1} \frac{2 t}{1-t^{2}}=\tan ^{-1} \frac{3 t-t^{3}}{1-3 t^{2}}, t \in(-\infty,-1) \cup\left(-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \cup(1, \infty) \\ &=\pi+\tan ^{-1} \frac{3 t-t^{3}}{1-3 t^{2}}, t \in\left(\frac{1}{\sqrt{3}}, 1\right) \\ &=-\pi+\tan ^{-1} \frac{3 t-t^{3}}{1-3 t^{2}}, t \in\left(-1,-\frac{1}{\sqrt{3}}\right) \end{aligned}

Step-by-Step Solution

Answer

These are a series of mathematical identities involving the inverse trigonometric functions and their properties. Once you understand how the inverse functions work and how to apply trigonometric identities, the solutions can be deduced straightforwardly. However, solving these identities requires knowledge of trigonometric formulas and some algebraic manipulation.

Step 1: Understand Inverse Trigonometric Functions

The inverse trigonometric functions are actually the reverse of the trigonometric functions. For example, if \(y = \sin(x)\), then \(x = \sin^{-1}(y)\). Then, using the properties of these functions, we can derive several identities.

Step 2: Apply Inverse Trigonometric Properties For Addition

Use the addition and subtraction formulas for the trigonometric functions. For example, the formula for the sum of two angles in the sine function states that \(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\). Now, we need to apply this formula to the inverse trigonometric functions.

Step 3: Work out The First Example as a Reference

For example, in proving that \(\sin^{-1}(3/5) + \sin^{-1}(8/17) = \sin^{-1}(77/85)\), first calculate the sines of the sum on the left-hand side using the sum of angles formula to get \(\sin(\sin^{-1}(3/5) + \sin^{-1}(8/17)) = \sin(\sin^{-1}(3/5))\cos(\sin^{-1}(8/17)) + \cos(\sin^{-1}(3/5))\sin(\sin^{-1}(8/17))\). Since we know that \(\sin(\sin^{-1}x) = x\) and \(\cos(\sin^{-1}x) = \sqrt{1 - x^2}\), plugging these values gives \((3/5) \sqrt{1 - (8/17)^2} + (8/17) \sqrt{1 - (3/5)^2}= 77/85\)

Step 4: Apply the Similar Steps for Other Parts

We can apply similar steps to prove the other identities.

Key Concepts

Trigonometric identities

Trigonometric identities are fundamental tools in solving trigonometric equations and simplifying expressions. They relate various trigonometric functions to one another. Some common trigonometric identities are the Pythagorean identities, such as

• ewline \( \begin{aligned} &\sin^2(\theta) + \cos^2(\theta) = 1, \ &1+\tan^2(\theta)=\sec^2(\theta), \ &1+\cot^2(\theta)=\csc^2(\theta). \end{aligned} \)

These identities arise from the geometric relations within a right-angled triangle.
Another key identity is the sum and difference identities, which help find the sine, cosine, or tangent of the sum or difference of two angles. For sine, it is written as:

• \(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\).

These identities are vital when dealing with inverse trigonometric functions, as they allow us to solve complex equations by breaking them down into simpler parts.

Inverse function properties

Inverse trigonometric functions such as \(\sin^{-1}, \cos^{-1}, \tan^{-1},\) and others, serve as the 'reverse' of the respective trigonometric function. These functions provide the angle whose trigonometric function gives a specific value. For example, if \(y = \sin(x)\), then \(x = \sin^{-1}(y)\). The properties of these inverse functions are essential to understand their behavior:

• Range: Each inverse trigonometric function has a specific range. For instance, \(\sin^{-1}(y)\) ranges between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
• Domain: The domain indicates the possible values for which these functions are defined. \(\sin^{-1}(x)\) is defined for \(-1 \leq x \leq 1\).
• Odd and Even Functions: \(\tan^{-1}(-x) = -\tan^{-1}(x)\) denotes it as an odd function, whereas \(\cos^{-1}(-x) = \pi - \cos^{-1}(x)\) highlights its unique behavior.

In the inverse functions, identities like \(\sin(\sin^{-1}(y)) = y\) and corresponding identities for other functions simplify complex calculations. They allow us to convert inverse trigonometric expressions into simpler algebraic forms.

Sum and difference formulas

The sum and difference formulas in trigonometry are powerful tools that relate the trigonometric functions of a sum or difference of angles to the functions of the individual angles. They have applications in various problems involving trigonometric functions and their inverses. For example, the sum formula for sine is:
\[\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\]
The cosine sum formula is slightly different and might look like this:

• \(\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\)

These formulas are extensively used in the context of inverse trigonometric functions to combine or separate angles effectively. When working with inverse functions, you often apply these formulas in the reverse way to retrieve compound angles.
To illustrate, if you need to prove \(\sin^{-1}(3/5) + \sin^{-1}(8/17) = \sin^{-1}(77/85)\), you'd calculate the sine of each part using the addition formula, then simplify. You rely on properties such as \(\sin(\sin^{-1}x) = x\) to express the result accurately. These transformations and evaluations allow us to break down and solve intricate trigonometric equations effectively.