 Hello and welcome to this session. In this session, let us discuss some properties of inverse trigonometric functions. Now we know that if sin theta is equal to x, then theta is equal to sin inverse x. Thus, we can see that sin inverse x is a symbol which denotes an angle or a number, the value of whose sin is x. So, the expressions sin inverse x, then cos inverse x, tan inverse x and so on are called inverse trigonometric functions. And now let us discuss some of the properties of inverse trigonometric functions. First of all, let us discuss the conversion property. The conversion property is sin inverse x is equal to cos inverse 1 minus x square which is further equal to tan inverse square root of 1 minus x square the whole which is further equal to inverse of 1 root of 1 minus x square the whole which is further equal to cos inverse of square root of 1 minus x square whole upon x the whole which is further equal to sin inverse 1 by x. Now let us start with its proof. Let sin inverse x is equal to theta which implies sin theta is equal to x. Now we know one of the trigonometric functions which is sin square theta plus cos square theta is equal to 1 and from that we have cos theta is equal to square root of 1 minus square theta which further implies cos theta is equal to square root of 1 minus. Now putting the value of sin theta it will be square root of 1 minus and this further implies theta is equal to cos inverse of square root of 1 minus now theta is equal to sin inverse x and here this is the value of theta this means sin inverse x is equal to cos inverse of square root of 1 minus x. Now tan theta is equal to cos theta which is further equal to we have taken sin theta as x and we are getting 1 minus x square theta is equal to tan inverse over square root of 1 minus x square of the whole. Now we know that sin theta is equal to 1 over cos theta which is further equal to 1 over cos theta is square root of 1 minus this implies theta is equal to sin inverse 1 over square root of theta is equal to 1 upon tan theta. Now over square root of 1 minus root of tan theta it will be square root of 1 minus x square implies theta is equal to root of 1 minus x square whole upon x is equal to x is equal to 1 upon sin theta which is written theta will be equal to 1 by x which further implies theta is equal to cos inverse 1 by x. Now these are the different values which we are getting are the values of theta that means we have proved the first property. We have to move conversion properties is equal to sin inverse square root of 1 minus x square is equal to tan inverse of square root of 1 minus x square over x the whole is equal to cosecant inverse square root of 1 minus x square the whole is equal to cot inverse root of 1 minus x square the whole is equal to secant inverse of 1 over x And the third parameter is that inverse x is equal to sine inverse of x over square root of 1 plus x square. The whole is equal to cos inverse square root of 1 plus x square the whole is equal to cosecant of 1 plus x square whole upon x the whole is equal to secant inverse of square root of 1 plus x square the whole is equal to root the first property. Similarly we can prove these two properties also. Now let us discuss the next property of inverse symmetric functions is equal to into square root of 1 minus x square plus x is equal to theta which implies x is equal to theta is equal to equal to now sine theta 1 minus sine square theta. Now we have got the value of sine 2 theta which further implies 2 theta is equal to sine inverse the whole which further implies now theta is equal to sine inverse x. So it would be 2 is equal to sine inverse of 2x into square root of 1 minus x square the whole. We have proved this property inverse x is equal to minus 1 the whole. Now let us prove this let cos inverse x is equal to theta which implies x is equal to cos theta theta minus 1 equal to now cos theta is x so it would be 2x square minus 1 which further implies 2 theta is equal to cos inverse of 2x square minus 1 the whole which implies now theta is equal to cos inverse x so it would be 2 is equal to 2x square minus 1 the whole. Now the next property is 3 sine inverse x is equal to sine inverse of 3x minus 4x cube the whole. Now let us start with its true let sine inverse x is equal to theta which implies x is equal to sine theta now up is equal to 3 sine theta minus 4 sine cube theta which implies sine 3 theta is equal to now sine theta is x so it is 3x minus 3 theta that is 3 and theta is sine inverse x is equal to sine inverse of 3x minus 4x cube the whole. Now let us prove the next property which is 3 cos inverse x is equal to cos inverse of 4x cube minus 3x the whole. Now let us start with its proof now here we know that cos 3 theta is equal to 4 minus 3 cos theta theta is equal to 4 now cos theta we have taken as x so this is equal to 4 into x cube minus 3 over theta that is 3 and theta is cos inverse x is equal to cos inverse of 4x cube minus 3x the whole. 3 tan inverse x is equal to tan inverse of 3x minus x cube whole upon 1 minus 3x square the whole. Now let us prove now let x is equal to theta which implies x is equal to tan theta now we know that is equal to now putting the value of tan theta this implies 3 theta is equal to 3x minus x cube whole upon 1 minus 3 which further implies now theta is tan inverse x so it will be 3 tan inverse x is equal to 3x minus x cube 3x square the whole. Let us discuss the next property which is y is equal to sine inverse to square root of 1 minus y square into square root of 1 minus x square the whole. If theta is equal to 0 y square is less than equal to 1. Now to this prove let sine inverse x is equal to alpha minus y is equal to beta and this implies minus beta the whole is equal to sine alpha cos alpha. So here we will change for in terms of address we will express cos beta and cos alpha in terms of sine so that we can get this result. Now this will be equal to now we know that by reciprocal identity that sine square beta plus cos square beta is equal to 1 therefore cos beta is equal to square root of 1 minus square beta plus minus square root of 1 minus sine square alpha beta. Now putting the sine beta plus minus beta the whole is equal to x into square root of 1 minus y square square into plus y is equal to sine inverse x into square root of 1 minus into square root of 1 minus x square the whole. So we have proved this property. Now the next property is y is equal to pi minus sine inverse minus y square i into square root of 1 minus x square or greater than equal to 0 square plus y square is greater than 1. Now let us start with this prove y is greater than 0 y square is greater than 1 x into square root of 1 minus y square plus minus pi into square root of square root of y and inverse y dot 1 minus x square the whole. This is the next property which is cos inverse x plus minus cos inverse y xy minus plus square root of 1 minus x square into square root of 1 minus y square the whole if xy are greater than 0 and x square this y square is less than equal to 1. Now let us start with this prove now let cos inverse x is equal to alpha y is equal to beta which implies x is equal to cos alpha and y is equal to cos beta. Now by using the formula plus minus beta the whole is equal to cos alpha cos beta minus plus minus beta the whole is equal to cos alpha into cos beta. Now using the trigonometrical identity sine alpha will be equal to square root of 1 minus cos square alpha into sine beta will be equal to square root of 1 minus cos square beta. Now this further implies minus beta the whole is equal to cos equal to x and cos beta is equal to y therefore this will be x into y minus plus square root of into square root of 1 minus y square. Now alpha is equal to cos inverse x and beta is equal to cos inverse y minus beta x plus minus cos inverse y which is equal to of xy minus plus square root of 1 minus x square into square root of 1 minus y square the whole. So we have proved the property. Now the next property is inverse y is equal to pi minus cos inverse of x square into square root of 1 minus y square the whole x plus minus sine inverse y that is this property we can prove this property also. So in this session we have learnt about some properties of inverse trigonometric functions and this completes our session hope you all have enjoyed this session. .