Fundamental Identities

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Take cos theta, square it, then take the square root. You get cos theta back, right? Test it with theta equal to pi. Cos pi is minus one, its square is one, and the square root of one is one. So where did the minus sign go? Hold that thought. Today we learn the fundamental identities, which come up often in JEE questions. So, let's begin.

0:30The three Pythagorean identities

Here is the unit circle, a circle of radius one. The Greek letter theta is our angle, and the point P has coordinates cos theta and sine theta. So the green side is cos theta, the blue side is sine theta, and the orange radius is one. What does Pythagoras give you? Yes, exactly! Sine squared theta plus cos squared theta equals one. The little two written up high means sine theta, the whole squared. And P stays on this circle for every angle, so this holds for every theta. Okay? Now divide both sides by cos squared theta. What do you get? One plus tan squared theta equals sec squared theta, very good! Sine over cos is tan, and one over cos is sec, all squared. Right? In the same way, divide by sine squared theta, and one plus cot squared theta equals cosec squared theta. Clear? But careful, we divided. So the second needs cos theta not equal to zero, and the third needs sine theta not equal to zero; the equals sign with a slash means not equal. These identities hold wherever their terms are defined. Got it?

1:43Consequences worth remembering

Now some time savers. Sec squared theta minus tan squared theta is one, and it factorises as a difference of squares. So sec theta plus tan theta equals one over, sec theta minus tan theta. They are reciprocals! And cosec theta plus cot theta equals one over, cosec theta minus cot theta. Okay? Now higher powers; the four written up high means the fourth power. How would you get sine to the four theta plus cos to the four theta from our first identity? Square both sides, that is correct! The square of the sum has an extra middle term, two sine squared cos squared. So sine to the four theta plus cos to the four theta equals one, minus two sine squared theta cos squared theta. Right? For the sixth powers, where the six up high means the sixth power, cube both sides instead. Sine to the six theta plus cos to the six theta equals one, minus three sine squared theta cos squared theta. Clear? Now, sine squared theta and cos squared theta are never negative, and they add up to one. Their product is largest when both are one half, so it lies from zero to one quarter; this sign means less than or equal to. Tell me, what is the least value of sine to the four plus cos to the four? One half, yes, exactly! One minus two quarters is one half, and the greatest is one. The curly E sign means belongs to, and the square brackets include both ends. For the sixth powers, one minus three quarters gives one quarter, so they lie from one quarter to one. Got it?

3:20How to prove identities, and a trap

Now, how do you prove an identity? There are three simple rules. Rule one: write everything in sine and cos, so tan, cot, sec and cosec disappear. Okay? Rule two: work on the more complicated side, and simplify it until it becomes the other side. Right? Rule three is smart: replace a one by sec squared theta minus tan squared theta, or by cosec squared theta minus cot squared theta. It often makes a factor that cancels. With me? To eliminate theta from two equations, solve for two ratios, then link them with a Pythagorean identity. Clear? Now, back to our puzzle. Is the square root of cos squared theta always equal to cos theta? No, not always, and that is the trap! Square roots are never negative. So the root of cos squared theta is mod cos theta; the vertical bars mean modulus, the size without the sign. At theta equal to pi, that gives one. That is where our minus sign went! So whenever you remove a root, check the sign from the quadrant. Got it?

4:24Topper's trick: least values

Now a topper's trick, where the letters a and b are greater than zero. The least value of a squared times tan squared theta, plus b squared times cot squared theta, is two times a times b. That works because tan times cot is one. Okay? And since sec squared is one plus tan squared, and cosec squared is one plus cot squared, the least value of a squared times sec squared theta, plus b squared times cosec squared theta, is a plus b, the whole squared. Clear?

4:51Example 1: prove (1 + cot - cosec)(1 + tan + sec) = 2

Alright, example one, from the course. Prove that one plus cot theta minus cosec theta, times one plus tan theta plus sec theta, equals two. Use rule one. Pause the video and try it first. Done? Okay, let's check. Cot is cos over sine, and cosec is one over sine. So the first bracket is sine theta plus cos theta minus one, over sine theta. Clear? Now you do the second bracket. What does it become? Cos theta plus sine theta plus one, over cos theta, yes, exactly! Tan is sine over cos, and sec is one over cos. Okay? Now look at the two numerators: sine plus cos, minus one, and sine plus cos, plus one. What do they remind you of? It is a difference of squares, correct! The product is sine theta plus cos theta, the whole squared, minus one, over sine theta cos theta. Right? Now expand. Sine squared plus cos squared is one, so the top is one, plus two sine theta cos theta, minus one. Two sine theta cos theta over sine theta cos theta is two. This holds where the expression is defined, with sine theta and cos theta not zero. Clear?

6:06Example 2: simplify root (1 - sin)/(1 + sin) in quadrant II

Now example two, a little harder. Simplify the square root of, one minus sine theta, over one plus sine theta, for theta between pi over two and pi. Pi is the Greek letter you see, and the two less-than signs mean theta lies strictly between them. Keep our puzzle in mind. Pause the video and try it first. Done? Okay, let's check. Multiply the top and the bottom by one minus sine theta. The top becomes one minus sine theta, the whole squared, and the bottom, one minus sine squared theta, is cos squared theta. With me? So the square root is simply one minus sine theta, over cos theta. That is sec theta minus tan theta, right? Oh no, careful, that is our puzzle again! The root of cos squared theta is mod cos theta. The top is safe: one minus sine theta is greater than or equal to zero, which is what this sign means. Clear? Theta lies between pi over two and pi, the second quadrant, where cos theta is less than zero, negative. So mod cos theta equals minus cos theta. Okay? So the root is one minus sine theta, over minus cos theta. That is minus sec theta plus tan theta, which is tan theta minus sec theta. Quick check: in the second quadrant, sec theta minus tan theta is negative, but a square root never is.

7:28Quick revision: three identities and the mod cos trap

Got it? So, quick revision. Sine squared theta plus cos squared theta equals one; divide by cos squared or sine squared for the other two, where defined. Sec theta plus tan theta is the reciprocal of sec theta minus tan theta, and sine to the four plus cos to the four lies from one half to one. And the root of cos squared theta is mod cos theta, so check the sign from the quadrant. Now try the practice problems on this page. All the best!