The CBI, the Nippon foundation, mathematics package for teachers title of the video secant, 90 degrees, minus theta and cosecant, 90 degrees, minus theta when theta equals 45 degrees, we have to prove that secant, 90 degrees, minus theta equals cosecant, theta and cosecant, 90 degrees, minus theta is secant theta that is secant, 45 degrees and cosecant, 45 degrees. Now let us take a right angle triangle with the three vertices, a b and c with the 90 degrees at the vertex b. Now, uh, we have taken the 45 degrees. As the reference angle, so an is theta, which is the 45 degrees. So when we take theta as the reference angle, then the side opposite to theta is the opposite side.

So that is bc is the opposite side. A b is the adjacent side and ac is the hypotenuse side. As the reference degree is 45. We have to find out the secant, 45 degrees, cosecant, 45 degrees.

And then the secant, 90 degrees, minus 45 degrees and cosecant, 90 degrees, minus 45 degrees that becomes again, 45 degrees, 45 degrees. So it is enough if we simply. Find out cosecant, 45 degrees and secant, 45 degrees that is nothing, but secant, 90 degrees, minus 45 degrees and cosecant, 90 degrees, minus 45 degrees. Now, when we have the theta as the reference angle, which is 45 degrees. Now the secant is defined as the hypotenuse divided by the adjacent when you have the 45 degrees, the sides of the triangle, satisfy some characteristics, let us measure a b and let us measure.

Bc. We can notice that when an is 45 degrees, then a b is one unit. B, c is one unit when a b and. B, c are one unit each then a c becomes square root of two one square, plus 2 square that is 2, take that square root that is square root of 2. Now the secant theta is hypotenuse divided by the adjacent.

So hypotenuse is square root of 2, divided by adjacent. That is 1. That is nothing but square root of 2. Now, cosecant theta is hypotenuse, divided by opposite. That is square root of 2, divided by 1. It is square root of 2. Now we have already proved that secant 90 degrees, minus 45 degrees equals 45.

Degrees that are again, square root of 2. Now if you just want to, you know, use c as the reference angle, then c becomes 90 minus 45 degrees. So that is 90 minus theta. So just for the practical purposes when you take c as the reference angle, then a b becomes the opposite and bc becomes the adjacent. So what is secant 90 minus 45 degrees when you take c as the reference that is hypotenuse by adjacent? What is hypotenuse?

It is square root of two. What is adjacent? It is 1. So that is square root of 2. What. Is cosecant 90 degrees, minus 45 degrees. That is if you apply the formula, it is hypotenuse divided by the opposite.

Now what is hypotenuse? It is square root of 2, divided by opposite. It is 1 that is square root of 2. So when you take 45 degrees, then you will be able to demonstrate that secant, 45 degrees equals cosecant, 45 degrees. And if you take the complementary angle, also, then it is the same because in the case of 45 degrees, the complementary angle becomes the angle itself in the earlier example, We used 30 degrees and 60 degrees for identifying the trigonometric ratios of complementary angles.

So here we have specifically taken 45 degrees to highlight that when the angle 45 degrees is taken the angle, and the complementary angles remain the same.