If - +1=0 prove that 1+ - Vidyadip

Question

If $\sin\alpha\sin\beta-\cos\alpha\cos\beta+1=0$ prove that $1+\cot\alpha\tan\beta$

✓

Answer

We have,
$\sin\alpha\sin\beta-\cos\alpha\cos\beta+1=0$
$\Rightarrow-(\cos\alpha\cos\beta-\sin\alpha\sin\beta)=-1$
$\Rightarrow-\cos(\alpha+\beta)=1\ ...(1)$
$\therefore\sin(\alpha+\beta)=\sqrt{1-\cos^2(\alpha+\beta)}$
$=\sqrt{1-1^2}=0$
$\Rightarrow\sin(\alpha+\beta)=0\ ...(2)$
Now,
$1+\cot\alpha\tan\beta=1+\frac{\cos\alpha}{\sin\alpha}\times\frac{\sin\beta}{\cos\beta}$
$=\frac{\sin\alpha\times\cos\beta+\cos\alpha\times\sin\beta}{\sin\alpha\times\cos\beta}$
$=\frac{\sin(\alpha+\beta)}{\sin\alpha\times\cos\beta}=\frac{0}{\sin\alpha\times\cos\beta}$ [Using equation (2)]
$\therefore1+\cot\alpha\tan\beta=0$
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 "5 Marks Questions" from Trigonometric Functions→Trigonometric Functions — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the standard deviation for the following data:

x	2	3	4	5	6	7
f	4	9	16	14	11	6

Sum the following series to n terms:
2 + 4 + 7 + 11 + 16 + ......

Solve the following equations:
$\tan\text{x}+\tan2\text{x}+\tan3\text{x}=0$

If $\sin\text{x}+\sin\cos\text{x}=\text{m},$ then prove that $\sin^6\text{x}+\cos^6\text{x}=\frac{4-3(\text{m}^2-1)^2}{4},$ where $\text{m}^2\leq2$

The frequency distribution:

x	A	2A	3A	4A	5A	6A
f	2	1	1	1	1	1

where A is a positive integer, has a variance of 160. Determine the value of A.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}\bigg\{\frac{\text{x}-2}{\text{x}^2-\text{x}}-\frac{1}{\text{x}^2-3\text{x}^2+2\text{x}}\bigg\}$

Solve the following systems of linear equations has no solution:
$\text{x}+2\text{y}\leq3,3\text{x}+4\text{y}\geq12,\text{x}\geq0,\text{y}\geq1.$

Prove that $\frac{\text{n}^{11}}{11}+\frac{\text{n}^5}{5}+\frac{\text{n}^3}{3}+\frac{62}{165}$ n is a positive integer for all $\text{n}\in\text{N}.$

Find the sum of the series whose n^th term is:

n(n + 1)(n + 4)

Find the area of the triangle formed by the lines:
y = 0, x = 2 and x + 2y = 3.