Q.Prove the following: If $\sin\alpha\sin\beta-\cos\alpha\cos\beta+1=0$ then prove $\cot\alpha\tan\beta=-1$

When two angles A and B are combined by addition or subtraction, the trigonometric ratio of the combined (compound) angle can be written in terms of the separate sine, cosine and tangent of A and B. The four building-block identities are $\cos(A-B)=\cos A\cos B+\sin A\sin B$, $\cos(A+B)=\cos A\cos B-\sin A\sin B$, $\sin(A+B)=\sin A\cos B+\cos A\sin B$ and $\sin(A-B)=\sin A\cos B-\cos A\sin B$, proved geometrically using the distance between two points on a unit circle. Dividing the sine formulas by the cosine formulas gives the tangent versions $\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}$, and taking reciprocals gives the cotangent versions. A very common use is to write an 'unfriendly' angle like $15°$, $75°$ or $105°$ as a sum or difference of the standard angles $30°,45°,60°,90°$ so its exact value can be found. The same identities let us prove identities that mix several angles, and, with $B=\dfrac{\pi}{2}$, they generate the allied-angle results.

"Compound angle formulas sin(A+B) cos(A-B) derivation" and "trigonometric functions class 11 important questions" are frequently searched around the Trigonometric Functions chapter of the NCERT/CBSE Class 11 Mathematics curriculum, since these four building-block identities are tested every year in board exams and JEE Main. Using them to evaluate "unfriendly" angles like $15°$ and $75°$, as shown here, is one of the most commonly repeated question types in this chapter.