Evaluate

Question

2cos^2(π/5)+2sin^2(π/5)? 

Also: 

Rewrite (1/(1+sinx)) so that it is not in fractional form. 

It would be great if an explanation and steps were provided. Thank you!

 

Answer 1

 

1)

 

The expression is 2cos² (π/5) + 2sin² (π/5).

Common out 2 from the expression.

= 2[cos² (π/5) + sin² (π/5)]

Pythagorean identity : sin² θ + cos² θ = 1.

= 2(1)

 = 2.

Therefore, 2cos² (π/5) + 2sin² (π/5) = 2.

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2)

The expression is 1/(1 + sin x).

Multiply numerator and denominator by (1 - sin x).

= (1 - sin x)/[(1 + sin x)(1 - sin x)]

= (1 - sin x)/(1 - sin² x)

Pythagorean identity : sin² θ + cos² θ = 1.

= (1 - sin x)/(cos² x)

= 1/cos² x - sin x/cos² x

= (1/cos x)² - [(sin x/cos x)*(1/cos x)]

Using reciprocal identity : 1/cos x = sec x.

= sec² x - [(sin x/cos x)*(sec x)]

Trigonometric identity : tan x = sin x/cos x.

= sec² x - [tan x sec x]

= sec x[sec x - tan x].

Therefore, 1/(1 + sin x) = sec x[sec x - tan x].