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Question

In fig 8.25, write sec a, tan c, cosec b, cot (90 - c), cos (90 - b) and find the value of sec2a - tan2 a.

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Solution

We have:sec a = hypotenuseadjacent side of a = SRTRtan c = opposite side of cadjacent side of c= SPPTcosec b = hypotenuseopposite side of b= TRTQ


In â–³STR, we have:
∠STR + ∠SRT + ∠TSR = 180o (Angle sum property of a triangle)
⇒ 90o + a + ∠ TSR = 180o
⇒ ∠ TSR = 90o − a
Similarly, ∠PST = 90o − c and ∠RTQ = 90o − b

Now, we have:cot (90°-a) = Adjacent side of (90°-a)Opposite side of (90°-a)=STTRsin (90°-c) = Opposite side of (90°-c)Hypotenuse= PTSTcos (90°-b) = Adjacent side of (90°-b)Hypotenuse=TQTRtan a = opposite side of aadjacent side of a= STTR


Now, consider sec2 a – tan2 a

= SRTR2 - STTR2= SR2-ST2TR2= TR2TR2 = 1 (Pythagoras theorem)

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