 Hi and welcome to the session. Today we will learn about properties of inverse trigonometric functions. First property is sin inverse of 1 upon x is equal to cosecant inverse x for x greater than equal to 1 or x less than equal to minus 1. Let's see its proof. We will assume cosecant inverse x is equal to y. So this gives us x is equal to cosecant y. Now let's take reciprocal on both the sides. So we get 1 upon x is equal to 1 upon cosecant y that is sin y. So this gives sin inverse of 1 upon x is equal to y. Now let's substitute the value of y that is cosecant inverse x. So we have sin inverse 1 upon x is equal to cosecant inverse x. So we have proved first property. Similarly we have second property that is cosecant inverse 1 upon x is equal to cosecant inverse x for x greater than equal to 1 or x less than equal to minus 1. And there is third property that is tan inverse 1 upon x is equal to cosecant inverse x for x greater than 0. Similarly we can prove these two properties as we did for this. Now let's move on to second set of properties. Here first property is sin inverse of minus x is equal to minus sin inverse x where x belongs to the closed interval minus 1, 1. Let's see the proof for this one. Here let's assume that sin inverse of minus x is equal to y. This implies minus x is equal to sin y. Now multiplying both sides by minus 1 we will get x is equal to minus sin y which is equal to sin of minus y. As we know the sin of minus theta is equal to minus sin theta. Now this gives us sin inverse of x is equal to minus y. Now we will substitute the value of y that is sin inverse of minus x. So we get sin inverse of x is equal to minus sin inverse of minus x. So with this we prove the first property. Similarly we have second property that is tan inverse of minus x is equal to minus tan inverse of x for x belonging to r. And third property is cosecant inverse of minus x is equal to minus cosecant inverse of x for mod x greater than equal to 1. We can prove these two properties in the same method. Now we will move on to third set of properties in which the first property is cosecant inverse of minus x is equal to pi minus cosecant inverse of x for x belonging to the closed interval minus 1, 1. Let's see the proof for this one. Let us assume cosecant inverse of minus x is equal to y. So this implies minus x is equal to cosecant inverse of x. Now multiplying both sides by minus 1 we get x is equal to minus cosecant inverse of x which is equal to cosecant inverse of x. So cosecant inverse of x is equal to pi minus y. Now we will substitute the value of y so we get pi minus cosecant inverse of minus x. Therefore now let's take cosecant inverse of minus x to the left side and cosecant inverse x to the right side. So we get cosecant inverse of minus x is equal to pi minus cosecant inverse x and this proves the given property. Similarly we can prove the second property which states cosecant inverse of minus x is equal to pi minus cosecant inverse of x for modulus or mod x greater than equal to 1. And the third property cot inverse of minus x is equal to pi minus cot inverse of x where x belongs to the set of real numbers. After this we have fourth set of properties in which the first property is sine inverse x plus cos inverse x is equal to pi upon 2 for x belongs to the closed interval minus 1, 1. Let's prove this property. Let sine inverse x is equal to y. So this implies x is equal to sine y which is equal to cos of pi by 2 minus y. So this implies cos inverse of x is equal to pi by 2 minus y. Now let us substitute the value of y that is sine inverse x. So this is equal to pi by 2 minus sine inverse x. Therefore we get sine inverse x plus cos inverse x is equal to pi by 2. With this we prove the given property. Similarly we can prove the second property tan inverse x plus cot inverse x is equal to pi by 2 for x belonging to r. And third property is cosecant inverse x plus secant inverse x is equal to pi by 2 for modulus x greater than equal to 1. In the fifth set of properties first property is tan inverse x plus tan inverse y is equal to tan inverse x plus y upon 1 minus xy for x into y less than 1. The proof for this property is as follows. Let tan inverse x is equal to theta and tan inverse y is equal to phi. So this implies x is equal to tan theta and y is equal to tan phi. Now we know that tan theta plus phi is equal to tan theta plus tan phi upon 1 minus tan theta into tan phi. Which is equal to x plus y upon 1 minus xy substituting the values of tan theta and tan phi which is equal to x and y respectively. So this gives theta plus phi is equal to tan inverse x plus y upon 1 minus xy. Now substituting the values of theta and phi we get tan inverse x plus tan inverse y is equal to tan inverse x plus y upon 1 minus xy. Similarly we can prove the second property that is tan inverse x minus tan inverse y is equal to tan inverse x minus y upon 1 plus xy for x into y greater than minus 1. And the third property which states 2 tan inverse x is equal to tan inverse 2x upon 1 minus x square for modulus x less than 1. Now let's go to sixth and last set of properties. Here the first property is 2 times tan inverse x is equal to sin inverse 2x upon 1 plus x square for modulus x less than equal to 1. Let's prove this property. Let us assume tan inverse x is equal to y. So this implies x is equal to tan y. Now in inverse 2x upon 1 plus x square will be equal to sin inverse 2 times tan y upon 1 plus tan square y. Which will be equal to sin inverse of sin 2y which is equal to 2y. Now we will substitute the value of y that is tan inverse x. So this will be equal to 2 times tan inverse x. Therefore we have 2 times tan inverse x is equal to sin inverse 2x upon 1 plus x square. Similarly we have second property that is 2 times tan inverse x is equal to cos inverse 1 minus x square upon 1 plus x square. For x greater than equal to 0 and third property 2 times tan inverse x is equal to tan inverse of 2x upon 1 minus x square for minus 1 less than x less than 1. And we can prove these two properties in the same manner. So with this we finish this session. Hope you must have understood all the properties. Goodbye, take care and have a nice day.