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 "(Each question 4 marks)" from Trigonometric Ratios of Multiple and Submultiple Angles→Trigonometric Ratios of Multiple and Submultiple Angles — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equation of the parabola whose: Focus is (3, 0) and the directrix is 3 x + 4y = 1

Prove that: $\sin3\text{A}+\sin2\text{A}-\sin\text{A}=4\sin\text{A}\cos\frac{\text{A}}{2}\cos\frac{3\text{A}}{2}$

Sketch the graphs of the following curves on the same scale and the same axes: $\text{y}=\cos^2\text{x}$ and $\text{y}=\cos\text{x}$

One diameter of the circle circumscribing the rectangle ABCD is 4y = x + 7. If the coordinates of Aand B are (-3, 4) and (5, 4) respectively, find the equation of the circle.

If the image of the point (2, 1) with respect to the line mirror be (5, 2), find the equation of the mirror.

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow-\infty}\big(\sqrt{4\text{x}^2-7\text{x}}+2\text{x}\big)$

Prove that:$\cos^2\text{A}+\cos^2\text{B}-2\cos\text{A}\cos\text{B}\cos\text{(A}-\text{B)}=\sin^2\text{(A}+\text{B)} $

Prove the following identities: $\frac{2\sin\text{x}\cos\text{x}-\cos\text{x}}{1-\sin\text{x}+\sin^2\text{x}-\cos^2\text{x}}=\cot\text{x}$

Prove the following by the principle of mathematical induction: $1^2 + 2^2 + 3^2 +...+ \text{n}^2=\frac{\text{n}(\text{n}+1)(2\text{n}+1)}{6}$

The variance of 20 observation is 5. If each observation is multiplied by 2, find the variance of the resulting observation.