Question

Solve question no 1

Answer 1

Step-by-step explanation:  

secteta = x+(1/4x) {equation 1 }  

Squaring both sides  

= sec²teta = [x+(1/4x)]²  

= sec²teta = (x)² + [(1/4 x)]² + 2.x.(1/4x) {because (a+b)² = (a)² + (b)² + 2ab }  

= sec²teta= x² + 1/16x² + 1/2  

Since sec² teta = 1 + tan² teta,  

So,  

1+ tan² teta= x² + 1/16x² + 1/2  

= tan² teta= x² + 1/16x² + 1/2 - 1  

= tan²teta = x² + 1/16x² + (1-2/2)  

= tan² teta = x² + 1/16x² - 1/2  

= (tan teta)² = [x-(1/4x)]² {because: (a)² + (b)² - 2ab = (a-b)² }  

=> tan teta = ± [x-(1/4x)] {equation 2 }  

Adding equation 1 and equation 2:  

= secteta+ tan teta = [x+(1/4x)] ± [ x-(1/4x)]  

IF IT IS +[ x-(1/4x)] THEN,  

sec teta + tan teta = (x+1/4x)+(x-1/4x)  

sec teta + tan teta= x + 1/4x + x - 1/4x  

sec teta + tan teta = 2x {equation I }  

IF IT IS -[ x-(1/4x)]  

sec teta + tan teta = (x+1/4x) - (x-1/4x)  

sec teta+ tan teta= x+1/4x - x + 1/4x  

sec teta + tanteta = 2/4x  

sec teta + tan teta = 1/2x {equation II }  

From equation I and equation II,  

secteta + tanteta = 2x or 1/2x  

HENCE, PROVED

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