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Evaluate:(i) $\frac{\sin ^{2} 63^{\circ}+\sin ^{2} 27^{\circ}}{\cos ^{2} 17^{\circ}+\cos ^{2} 73^{\circ}}$(ii) $\sin 25^{\circ} \cos 65^{\circ}+\cos 25^{\circ} \sin 65^{\circ}$

Precalculus

Chapter 8

Introduction to Trigonometry

Section 4

Trigonometric Ratios of Complementary Angles

Trigonometry

Piedmont College

University of Michigan - Ann Arbor

Idaho State University

Utica College

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So in the given question, we have a few trick no metric Expressions that we need to evaluate. So the first expression has given us science create 63° plus Science square 27 degrees divided by cost square 17. The grease Cost where 17° plus Car Square 73°.. So what we are going to do is we can use an identity that scientific data as it will do. Can we return as cost of 99 S Tita. And you can also use the identity that cost data can be written us sign of 90 minus theta. And we can use another property over here. Which is cost where theater plus sine square theater Is equal to one. So these are the identities that we are going to use over here. So first, what we do is we can write Science square 63 in terms of cost data by changing signed 63 As costs 90 -63. Right? And this is squared bless Science squared 27 and similarly in the denominator we can change Cause17 that has caused 17 squared as a sign of 90 -17 squared bless Costs Square 73. So what would happen is we would have Cause1963 is 27 plus Sine Squared 27. And in the denominator we will have 90 minus 17 is equal to 73. So we will have science square 73 plus cost square 73. So now since we know that cost with the tabla science where data is equal to one, the value of this numerator and denominator will be one. So one by one would give us the required value of this expression as one. Now let's move on to the next exhibition that is given in the question. So the next expression has given us signed 25 times cost 65 less cost 65 cost 25 times same 65. So in the same way What we can do over here is instead of sign 25 let's right cause of 90 -25 Times cost 65. Place over here. We can write instead of because we can write sign. So scores of 25 can build trust, signed off 90 -25 Times Sine 65. So then what we would have is a sign of cost of 90 -25 years, cost 65 degrees times costs 65 degrees would make it cost square 65 plus we have signs of 90 minus 25 years 65. So Science 65 times Sine 65 would be Science square 65. And now it is of the form of the identity cost where the tablet science where later Which would evaluate to one. So this is the value of the given techno metric expressions. I hope you understood the method. Thank you

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