MHT CET · Maths · Trigonometric Equations
If \(\alpha+\beta=\frac{\pi}{2}\) and \(\beta+\gamma=\alpha\), then \(\tan \alpha\) equals
- A \(2(\tan \beta+\tan \gamma)\)
- B \(\tan \beta+\tan \gamma\)
- C \(\tan \beta+2 \tan \gamma\)
- D \(2 \tan \beta+\tan \gamma\)
Answer & Solution
Correct Answer
(C) \(\tan \beta+2 \tan \gamma\)
Step-by-step Solution
Detailed explanation
Given, \(\alpha=\beta+\gamma\)
\(\therefore \gamma=\alpha-\beta \)
\( \tan \gamma =\tan (\alpha-\beta) \)
\( =\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \cdot \tan \beta}\)
\(=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \cdot \tan \left(\frac{\pi}{2}-\alpha\right)} \cdots[\alpha+\beta=\frac{\pi}{2} \)
\( \therefore \beta=\frac{\pi}{2}-\alpha] \)
\( =\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \cot \alpha} \)
\( =\frac{\tan \alpha-\tan \beta}{2}\)
\(\Rightarrow 2 \tan \gamma=\tan \alpha-\tan \beta\)
\(\Rightarrow \tan \alpha=2 \tan \gamma+\tan \beta\)
\(\therefore \gamma=\alpha-\beta \)
\( \tan \gamma =\tan (\alpha-\beta) \)
\( =\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \cdot \tan \beta}\)
\(=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \cdot \tan \left(\frac{\pi}{2}-\alpha\right)} \cdots[\alpha+\beta=\frac{\pi}{2} \)
\( \therefore \beta=\frac{\pi}{2}-\alpha] \)
\( =\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \cot \alpha} \)
\( =\frac{\tan \alpha-\tan \beta}{2}\)
\(\Rightarrow 2 \tan \gamma=\tan \alpha-\tan \beta\)
\(\Rightarrow \tan \alpha=2 \tan \gamma+\tan \beta\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- \(\int \frac{x^{2}+1}{(x-3)(x-2)} d x=P x+Q \log |x-3|+R \log |x-2|+c\), where \(c\) is constant of
integration, then the values of \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) are, respectivelyMHT CET 2020 Medium - The shortest distance between the lines \(1+x=2 y=-12 z\) and
\(x=y+2=6 z-6\) isMHT CET 2020 Easy - If \(\mathrm{I}=\int \frac{\mathrm{e}^x}{\mathrm{e}^{4 x}+\mathrm{e}^{2 x}+1} \mathrm{~d} x\) and \(\mathrm{J}=\int \frac{\mathrm{e}^{-x}}{\mathrm{e}^{-4 x}+\mathrm{e}^{-2 x}+1} \mathrm{~d} x\) then for any arbitrary constant c, the value of \(\mathrm{J}-\mathrm{I}\) equalsMHT CET 2023 Medium
- The number of common tangents that can be drawn to the circles \(x^2+y^2-6 x=0\) and \(x^2+y^2+6 x+2 y+1=0\) is .....MHT CET 2025 Medium
- The element in the third row and second column of the inverse of the \(\operatorname{matrix}\left[\begin{array}{ccc}3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 2 & 5\end{array}\right]\) isMHT CET 2022 Easy
- \(\mathrm{f}(x)= \begin{cases}{\left[x^2\right]-\left[-x^2\right],} & x \neq 3 \\ \mathrm{k} & , x=3\end{cases}\)
is continuous at \(x=3\), then \(\mathrm{k}=\) where \([\cdot]\) is greatest integer functionMHT CET 2025 Medium
More PYQs from MHT CET
- If \(y=\tan ^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), 0 \leq x < \frac{\pi}{2}\), then \(\frac{d y}{d x}\) at \(x=\frac{\pi}{6}\) isMHT CET 2021 Medium
- Which of the following polymers is obtained from \(\varepsilon\)-caprolactum?MHT CET 2021 Easy
- Identify the product ' B ' in the following series of reactions.
\(\text { Chlorobenzene } \frac{\text { i) } \mathrm{NaOH}, 623 \mathrm{~K} / 150 \mathrm{~atm}}{\text { ii) } \mathrm{H}_3 \mathrm{O}^{+}} \mathrm{A} \xrightarrow{\mathrm{Br}_2 \text { water }} \mathrm{B}\)MHT CET 2025 Medium - Molal elevation constant is the elevation in boiling point produced byMHT CET 2020 Easy
- Which of the following cannot be used as standard solution for determination of cell constant of conductivity cell?MHT CET 2024 Medium
- Two radioactive substances \(A\) and \(B\) have decay constants ' \(5 \lambda\) ' and ' \(\lambda\) ' respectively. At \(\mathrm{t}=0\), they have the same number of nuclei. The ratio of number of nuclei of A to those of B will be \(\left(\frac{1}{\mathrm{e}}\right)^2\) after a time intervalMHT CET 2024 Medium