Download App

Trigonometric Functions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Functions4 Marks
Question
If $\text{a}=\sec\text{x}-\tan\text{x}$ and $\text{b}=\text{cosec x}+\cot\text{x},$ then show that $\text{ab}+\text{a} - \text{b}+ 1=0.$
✓

Answer

$\text{L.H.S}=\text{ab + a} - \text{b + }1$ $=(\sec\text{x}-\tan\text{x})(\text{cosec+}\cot)+\sec\text{x}-\tan\text{x}-\text{cosec }\text{x}-\cot\text{x}+1$ $=\Big(\frac{1}{\cos\text{x}}-\frac{\sin\text{x}}{\cos\text{x}}\Big)\Big(\frac{1}{\sin\text{x}}+\frac{\cos\text{x}}{\sin\text{x}}\Big)+\frac{1}{\cos\text{x}}-\frac{\sin\text{x}}{\cos\text{x}}-\frac{1}{\sin\text{x}}-\frac{\cos\text{x}}{\sin\text{x}}+1$ $=\frac{1}{\sin\text{x}\cos\text{x}}+\frac{1}{\cos\text{x}}\times\frac{\cos\text{x}}{\sin\text{x}}-\frac{\sin\text{x}}{\cos\text{x}}\\\ \ \ \times\frac{1}{\sin\text{x}}-\tan\text{x}\times\cot\text{x}+\frac{1}{\cos\text{x}}-\frac{\sin\text{x}}{\cos\text{x}}-\frac{1}{\sin\text{x}}$ $=\frac{1}{\sin\text{x}\cos\text{x}}+\frac{1}{\sin\text{x}}-\frac{1}{\cos\text{x}}-1+\frac{1}{\cos\text{x}}-\frac{\sin\text{x}}{\cos\text{x}}-\frac{1}{\sin\text{x}}-\frac{\cos\text{x}}{\sin\text{x}}+1$ $=\frac{1}{\sin\text{x}\cos\text{x}}-\frac{\sin\text{x}}{\cos\text{x}}-\frac{\cos\text{x}}{\sin\text{x}}$ $=\frac{1\sin^2\text{x}-\cos^2\text{x}}{\sin\text{x}.\cos\text{x}}$ $=\frac{1-(\cos^2\text{x}+\sin^2\text{x})}{\sin\text{x}.\cos\text{x}}$ $=\frac{1-1}{\sin\text{x}.\cos\text{x}}=0$ $=\text{R.H.S. Hence Proved}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 4 marks)" from Trigonometric Functions→Trigonometric Functions — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove the following by using the principle of mathematical induction for all n ∈ N:$\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{\text{n(n+1)(n+2)}}=\frac{\text{n(n+3)}}{4(\text{n+1})(\text{n}+2)}.$
→
The mean and standard deviation of 20 observation are found to be 10 and 2 respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in cases of it is replaced by 12 .
→
If a, b, c are in A.P., prove that: $\text{a}^3+\text{c}^3+6\text{abc}=8\text{b}^3$
→
If A = {2, 3}, B = {4, 5}, C = {5, 6}, find $\text{A}\times(\text{B}\cap\text{C}),\text{ A}\times(\text{B}\cap\text{C}),(\text{A}\times\text{B})\cup(\text{A}\times\text{C}).$
→
Find the equation of the ellipse in the following case:focus is (0, 1), directrix is x + y = 0 and $\text{e}=\frac{1}{2}.$
→
For any two complex numbers $z_1$ and $z_2$, prove that $\operatorname{Re}\left(z_1 z_2\right)=\operatorname{Rez} z_1 \operatorname{Rez} z_2-\operatorname{Im} z_1 \mid m z_2$.
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow-1}\frac{\text{x}^2-\text{x}-2}{\big(\text{x}^2+\text{x}\big)+\sin(\text{x}+1)}$
→
Find the mean variance and standard deviation for the following data: $6, 7, 10, 12, 13, 4, 8, 12.$
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{{\pi}}}\frac{1+\cos\text{x}}{\tan^2\text{ x}}$
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{{\pi}}}\frac{\sqrt{2+\cos\text{x}-1}}{(\pi-\text{x})^2}$
→