What is the general solution for sin^2(a)sec(a)=tan^2(a) ?

The general solution for sin^2(a)sec(a)=tan^2(a) is a=2pin,a=pi+2pin

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sin^2(a)sec(a)=tan^2(a)

Solution

sin^{2} (a) sec (a) = tan^{2} (a)

Solution

a = 2 πn, a = π + 2 πn

Degrees

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (a) sec (a) = tan^{2} (a)

Subtract

tan^{2} (a)

from both sides

sin^{2} (a) sec (a) - tan^{2} (a) = 0

Express with sin, cos

- tan^{2} (a) + sec (a) sin^{2} (a)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - (\frac{sin ( a )}{cos ( a )})^{2} + sec (a) sin^{2} (a)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= - (\frac{sin ( a )}{cos ( a )})^{2} + \frac{1}{cos ( a )} sin^{2} (a)

Simplify

- (\frac{sin ( a )}{cos ( a )})^{2} + \frac{1}{cos ( a )} sin^{2} (a) : \frac{- sin ^{2} ( a ) + sin ^{2} ( a ) cos ( a )}{cos ^{2} ( a )}

- (\frac{sin ( a )}{cos ( a )})^{2} + \frac{1}{cos ( a )} sin^{2} (a)

(\frac{sin ( a )}{cos ( a )})^{2} = \frac{sin ^{2} ( a )}{cos ^{2} ( a )}

(\frac{sin ( a )}{cos ( a )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{sin ^{2} ( a )}{cos ^{2} ( a )}

\frac{1}{cos ( a )} sin^{2} (a) = \frac{sin ^{2} ( a )}{cos ( a )}

\frac{1}{cos ( a )} sin^{2} (a)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot sin ^{2} ( a )}{cos ( a )}

Multiply:

1 \cdot sin^{2} (a) = sin^{2} (a)

= \frac{sin ^{2} ( a )}{cos ( a )}

= - \frac{sin ^{2} ( a )}{cos ^{2} ( a )} + \frac{sin ^{2} ( a )}{cos ( a )}

Least Common Multiplier of

cos^{2} (a), cos (a) : cos^{2} (a)

cos^{2} (a), cos (a)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos^{2} (a)

cos (a)

= cos^{2} (a)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

cos^{2} (a)

For

\frac{sin ^{2} ( a )}{cos ( a )} :

multiply the denominator and numerator by

cos (a)

\frac{sin ^{2} ( a )}{cos ( a )} = \frac{sin ^{2} ( a ) cos ( a )}{cos ( a ) cos ( a )} = \frac{sin ^{2} ( a ) cos ( a )}{cos ^{2} ( a )}

= - \frac{sin ^{2} ( a )}{cos ^{2} ( a )} + \frac{sin ^{2} ( a ) cos ( a )}{cos ^{2} ( a )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- sin ^{2} ( a ) + sin ^{2} ( a ) cos ( a )}{cos ^{2} ( a )}

= \frac{- sin ^{2} ( a ) + sin ^{2} ( a ) cos ( a )}{cos ^{2} ( a )}

\frac{- sin ^{2} ( a ) + cos ( a ) sin ^{2} ( a )}{cos ^{2} ( a )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- sin^{2} (a) + cos (a) sin^{2} (a) = 0

Factor

- sin^{2} (a) + cos (a) sin^{2} (a) : sin^{2} (a) (cos (a) - 1)

- sin^{2} (a) + cos (a) sin^{2} (a)

Factor out common term

sin^{2} (a)

= sin^{2} (a) (- 1 + cos (a))

sin^{2} (a) (cos (a) - 1) = 0

Solving each part separately

sin^{2} (a) = 0 or cos (a) - 1 = 0

sin^{2} (a) = 0 : a = 2 πn, a = π + 2 πn

sin^{2} (a) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

sin (a) = 0

General solutions for

sin (a) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

Solve

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

cos (a) - 1 = 0 : a = 2 πn

cos (a) - 1 = 0

Move

1

to the right side

cos (a) - 1 = 0

Add

1

to both sides

cos (a) - 1 + 1 = 0 + 1

Simplify

cos (a) = 1

cos (a) = 1

General solutions for

cos (a) = 1

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = 0 + 2 πn

a = 0 + 2 πn

Solve

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

Combine all the solutions

a = 2 πn, a = π + 2 πn

Popular Examples

tan^2(θ)-sec(θ)=1

tan^{2} (θ) - sec (θ) = 1

sin(x)= 8/10

sin (x) = \frac{8}{10}

sin(x)cos(2x)+cos(x)sin(2x)=0

sin (x) cos (2 x) + cos (x) sin (2 x) = 0

7cos(5x)+9=12

7 cos (5 x) + 9 = 12

cos(2x)+1=sin(2x)

cos (2 x) + 1 = sin (2 x)

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(a)sec(a)=tan^2(a) ?
The general solution for sin^2(a)sec(a)=tan^2(a) is a=2pin,a=pi+2pin