If 𝜃 is in Quadrant II and cos𝜃+sin𝜃=1/5, the value of 𝑠𝑒𝑐𝜃/𝑡𝑎𝑛𝜃 + 𝑐𝑠𝑐𝜃/𝑐𝑜𝑡𝜃 would be (A) 3/5 (B) -5/12 (C)5/12 (D) 1. Ans: B

Answer 1

That is not calculus.

cos θ + sin θ = 1 / 5

Rise both sides to the power of two

( cos θ + sin θ )² = ( 1 / 5 )²

cos² θ + 2 sin θ cos θ + sin² θ = 1 / 25

2 sin θ cos θ + sin² θ + cos² θ = 1 / 25

2 sin θ cos θ + 1 = 1 / 25

Subtract 1 to both sides

2 sin θ cos θ = 1 / 25 - 1

2 sin θ cos θ = 1 / 25 - 25 / 25

2 sin θ cos θ = - 24 / 25

Multiply both sides by 1 / 2

sin θ cos θ = - 12 / 25

In Quadrant II sine is positive and cosine is negative so product of positive sine and negative cosine is negative.

This means:

sin θ cos θ = - 12 / 25 lie in Quadrant II

sec θ / tan θ + csc θ / cot θ = ( 1 / cos θ ) / tan θ + ( 1 / sin θ ) / cot θ =

[ 1 / ( cos θ ∙ tan θ ] + [ 1 / ( sin θ ∙ cot θ ] =

[ 1 / ( cos θ ∙ sin θ / cos θ ] + [ 1 / ( sin θ ∙ cos θ / sin θ ] =

1 / sin θ + 1 / cos θ = ( cos θ + sin θ ) / ( sin θ cos θ ) =

( 1 / 5 ) / ( - 12 / 25 ) = 1 ∙ 25 / 5 ∙ ( - 12 ) =

25 / 5 ∙ ( - 12 ) = 1 5cos θ + sin θ = 1 / 5

Rise both sides to the power of two

( cos θ + sin θ )² = ( 1 / 5 )²

cos² θ + 2 sin θ cosθ + sin² θ = 1 / 25

2 sin θ cosθ + sin² θ + cos² θ = 1 / 25

2 sin θ cosθ + 1 = 1 / 25

Subtract 1 to both sides

2 sin θ cosθ = 1 / 25 - 1

2 sin θ cosθ = 1 / 25 - 25 / 25

2 sin θ cosθ = - 24 / 25

Multiply both sides by 1 / 2

sin θ cosθ = - 12 / 25

In Quadrant II sine is positive and cosine is negative so product of positive sine and negative cosine is negative.

This means:

sin θ cosθ = - 12 / 25 lie in Quadrant II

sec θ / tan θ + csc θ / cot θ = ( 1 / cos θ ) / tan θ + ( 1 / sin θ ) / cot θ =

[ 1 / ( cos θ ∙ tan θ ] + [ 1 / ( sin θ ∙ cot θ ] =

[ 1 / ( cos θ ∙ sin θ / cos θ ] + [ 1 / ( sin θ ∙ cos θ / sin θ ] =

1 / sin θ + 1 / cos θ = ( cos θ + sin θ ) / ( sin θ cos θ ) =

( 1 / 5 ) / ( - 12 / 25 ) = 1 ∙ 25 / 5 ∙ ( - 12 ) = 5 ∙ 5 / 5 ∙ ( - 12 ) = - 5 / 12

Answer 2

One My typo.

Not

Not

[ 1 / ( cos θ ∙ sin θ / cos θ ] + [ 1 / ( sin θ ∙ cos θ / sin θ ] =

It should be like this

[ 1 / ( cos θ ∙ sin θ / cos θ ) ] + [ 1 / ( sin θ ∙ cos θ / sin θ ) ] =