MHT CET · Maths · Trigonometric Ratios & Identities
If \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\) are the angles of a cyclic quadrilateral taken in order, then \(\cos A+\cos B+\operatorname{Cos} C+\operatorname{Cos} D=\)
- A \(-1\)
- B 1
- C \(\frac{1}{2}\)
- D 0
Answer & Solution
Correct Answer
(D) 0
Step-by-step Solution
Detailed explanation
Since the quadrilateral \(\mathrm{ABCD}\) is cyclic, we have
\(\mathrm{A}+\mathrm{C}=180^{\circ}\) and \(\mathrm{B}+\mathrm{D}=180^{\circ}\)
\(\therefore \cos \mathrm{A}=\cos \left(180^{\circ}-\mathrm{C}\right)=-\cos \mathrm{C}\)
\(\cos \mathrm{B}=\cos \left(180^{\circ}-\mathrm{D}\right) \quad=-\cos \mathrm{D}\)
\(\therefore \cos \mathrm{A}+\cos \mathrm{B}+\cos \mathrm{C}+\cos \mathrm{D}=0\)
\(\mathrm{A}+\mathrm{C}=180^{\circ}\) and \(\mathrm{B}+\mathrm{D}=180^{\circ}\)
\(\therefore \cos \mathrm{A}=\cos \left(180^{\circ}-\mathrm{C}\right)=-\cos \mathrm{C}\)
\(\cos \mathrm{B}=\cos \left(180^{\circ}-\mathrm{D}\right) \quad=-\cos \mathrm{D}\)
\(\therefore \cos \mathrm{A}+\cos \mathrm{B}+\cos \mathrm{C}+\cos \mathrm{D}=0\)
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
- \(\int_0^1 \log (x+1) \mathrm{d} x=\)MHT CET 2025 Medium
- If the variance of the numbers \(-1,0,1, \mathrm{k}\) is 5 , where \(\mathrm{k}>0\), then \(\mathrm{k}\) is equal toMHT CET 2023 Easy
- \(\int \frac{x+\sin x}{1+\cos x} d x\) is equal toMHT CET 2007 Hard
- Let p and q be the position vectors of P and Q respectively, with respect to O and \(|\overline{\mathrm{p}}|=\mathrm{p},|\overline{\mathrm{q}}|=\mathrm{q}\). The points R and S divide PQ internally and externally in the ratio \(2: 3\) respectively. If \(O R\) and OS are perpendiculars, thenMHT CET 2024 Medium
- \(M\) and \(N\) are the mid points of the diagonals \(A C\) and \(B D\) respectively of quadrilateral \(A B C D,\) then
\(\overrightarrow{A B}+\overrightarrow{A D}+\overrightarrow{C B}+\overrightarrow{C D}=\)MHT CET 2016 Medium - The equation of the curve passing through the origin and satisfying the equation \(\left(1+x^2\right) \frac{\mathrm{dy}}{\mathrm{d} x}+2 x y=4 x^2\), isMHT CET 2025 Medium
More PYQs from MHT CET
- During how many steps Co-A is used in aerobic respiration?MHT CET 2024 Medium
- A series LCR circuit with resistance \(R=500 \mathrm{ohm}\) is connected to an a.c. source of \(250 \mathrm{~V}\). When only the capacitance is removed, the current lags behind the voltage by \(60^{\circ}\). When only the inductance is removed, the current leads the voltage by \(60^{\circ}\). The impedance of the circuit isMHT CET 2021 Medium
- Which among the following is a product of hydrolysis of one mole raffinose?MHT CET 2020 Easy
- What is the volume (in \(\mathrm{dm}^3\) ) occupied by \(75 \mathrm{~g}\) ethane at S.T.P.?MHT CET 2021 Easy
- As we go from the equator of the earth to pole of the earth, the value of acceleration due to gravityMHT CET 2020 Easy
- Which among the following halogen does not form polyhalide ion?MHT CET 2020 Medium